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added autodiff reverse docs
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mzhelyez
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doc/differentiation/autodiff_reverse.qbk
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doc/differentiation/autodiff_reverse.qbk
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[/ Copyright Maksym Zhelyeznyakov 2025-2026
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// Distributed under the Boost Software License, Version 1.0.
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// (See accompanying file LICENSE_1_0.txt or copy at
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// https://www.boost.org/LICENSE_1_0.txt)]
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[section:autodiff Reverse Mode Automatic Differentiation]
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[template autodiff_equation[name] '''<inlinemediaobject><imageobject><imagedata fileref="../equations/autodiff/'''[name]'''"></imagedata></imageobject></inlinemediaobject>''']
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[template autodiff_graph[name] '''<inlinemediaobject><imageobject><imagedata fileref="../graphs/autodiff/'''[name]'''"></imagedata></imageobject></inlinemediaobject>''']
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[h1:synopsis Synopsis]
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#include <boost/math/differentiation/autodiff_reverse.hpp>
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namespace boost {
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namespace math {
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namespace differentiation {
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namespace reverse_mode {
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/* autodiff variable of type RealType (numeric types), stores derivatives up to DerivativeOrder*/
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template<typename RealType, size_t DerivativeOrder = 1>
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class rvar : public expression<RealType, DerivativeOrder, rvar<RealType, DerivativeOrder>> {
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// inner_t of rvar<RealType, N> = var<RealType, N-1>
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// used to store graphs of N-1 order derivatives
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// rvar_t<RealType, 0> decays to RealType
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using inner_t = rvar_t<RealType, DerivativeOrder - 1>;
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/* constructors */
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// default
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rvar();
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// construct from numeric type
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rar(const RealType value);
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// copy
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rvar(const rvar<RealType, DerivativeOrder> other);
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/* assignment operators */
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// from numeric type
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rvar &operator=(RealType v);
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// from another rvar
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rvar &operator=(const rvar<RealType, DerivativeOrder> &other);
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// from a chain of expression templates
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rvar &operator=(const expression<RealType, DerivativeOrder, E> &expr);
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// calculate all derivatives in computational graph
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void backward();
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// get RealType value in rvar
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RealType item() const;
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// get accumulated adjoint of this variable
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// returns inner_t
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const inner_t &adjoint() const { return *node_->get_adjoint_ptr(); }
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inner_t &adjoint() { return *node_->get_adjoint_ptr(); }
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// all arithmetic and comparison operators are overloaded
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template<class E>
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rvar<RealType, DerivativeOrder> &operator+=(const expression<RealType, DerivativeOrder, E> &expr);
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template<class E>
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rvar<RealType, DerivativeOrder> &operator*=(const expression<RealType, DerivativeOrder, E> &expr)
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}
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// gradient tape holds the computational graph
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template<typename RealType, size_t DerivativeOrder, size_t buffer_size = BOOST_MATH_BUFFER_SIZE>
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class gradient_tape {
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// clear tape
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void clear();
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// sets all derivatives to zero
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void zero_grad();
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// adds a checkpoint you can rewind to
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void add_checkpoint();
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// rewinds tape to last checkpoint set
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void rewind_to_last_checkpoint();
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// index is "checkpoint" index. so // order which checkpoint was set
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void rewind_to_checkpoint_at(size_t index);
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// rewind to beginning of graph without clearing the tape
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void rewind();
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}
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// tape access
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template<typenametemplate<typename RealType, size_t DerivativeOrder>
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inline gradient_tape<RealType, DerivativeOrder, BOOST_MATH_BUFFER_SIZE> &get_active_tape();
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// standard math functions are overloaded using expression templates
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template<typename RealType, size_t DerivativeOrder, typename LHS, typename RHS>
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struct add_expr : public abstract_binary_expression<RealType, DerivativeOrder, LHS, RHS, add_expr<RealType, DerivativeOrder, LHS, RHS>>
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template<typename RealType, size_t DerivativeOrder, typename LHS, typename RHS> add_expr<RealType, DerivativeOrder, LHS, RHS> operator+(const expression<RealType, DerivativeOrder, LHS> &lhs,const expression<RealType, DerivativeOrder, RHS> &rhs);
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// Standard math functions are overloaded and called via argument-dependent lookup (ADL).
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template<typename RealType, size_t DerivativeOrder, typename ARG>
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floor_expr<RealType, DerivativeOrder, ARG> floor(const expression<RealType,DerivativeOrder, ARG> &arg);
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template<typename RealType, size_t DerivativeOrder, typename ARG>
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cos_expr<RealType, DerivativeOrder, ARG> cos(const expression<RealType,DerivativeOrder, ARG> &arg);
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// Helper gradient functions
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// rvar<T,0> decays into T
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// returns vector<rvar<T,order1_1-1>> of gradients
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template<typename T, size_t order_1, size_t order_2>
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auto grad(rvar<T, order_1> &f, std::vector<rvar<T, order_2> *> &x);
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template<typename T, size_t order_1, typename First, typename... Other>
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auto grad(rvar<T, order_1> &f, First first, Other... other)
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// returns a vector<vector<rvar<T, order_1 - 2>> representing the hessian matrix
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template<typename T, size_t order_1, size_t order_2>
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auto hess(rvar<T, order_1> &f, std::vector<rvar<T, order_2> *> &x)
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template<typename T, size_t order_1, typename First, typename... Other>
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auto hess(rvar<T, order_1> &f, First first, Other... other)
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// returns N-d nested vector<vector< ...rvar<T,order_1 - N> ...
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// representing the tensor generalization of the N-d gradient
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template<size_t N, typename T, size_t order_1, size_t order_2>
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auto grad_nd(rvar<T, order_1> &f, std::vector<rvar<T, order_2> *> &x)
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template<size_t N, typename ftype, typename First, typename... Other>
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auto grad_nd(ftype &f, First first, Other... other)
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// ...
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} // namespace reverse_mode
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} // namespace differentiation
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} // namespace math
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} // namespace boost
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[h1:description Description]
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Reverse mode autodiff is a header-only C++ library the [@https://en.wikipedia.org/wiki/Automatic_differentiation
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automatic differentiation] (reverse mode) of mathematical functions. This implementation builds a computational graph known as a tape, which sotres all operations and their corresponding derivatives. The total gradients are then computed by reverse accumulation via the chain rule.
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Consider the following function
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template<typename T>
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T f(x,y)
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{
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return x*y+sin(x);
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}
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int main()
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{
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rvar<double,1> x = 1.0;
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rvar<double,1> y = 1.0;
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rvar<double,1> z = f(x,y);
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z.backward();
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}
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and the associated computational graph.
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[:[:[autodiff_graph autodiff_reverse_graph.svg]]]
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Using reverse mode autodiff, the gradients are computed by traversing the graph backward:
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[:[:[autodiff_equation reverse_mode_autodiff_ex_eq.svg]]]
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Some key points about reverse mode automatic differentiation:
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1. Reverse mode auto-diff is exceptionally efficient for computing the gradient of functions mapping from high to low dimensional space, f : [real][super n][rarr][real]. Unlike finite differences or forward mode autodiff which scale with the number of input variables, reverse mode autodiff calculates the entire gradient vector in time proportional to the original function's evaluation time.
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2. It is possible to compute higher order derivatives by applying reverse mode over reverse mode differentiation. The backward function builds a new computational tape over the gradient computation. This approach is conceptually sound, but it can become computationally expensive very quickly. Forward mode autodiff is generally preferrable when computing higher order derivatives.
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3. While reverse mode is fast for computing the gradient, it has to store all the intermediate values from the forward pass to be used during the backward pass. This can be a significant memory overhead, especially for deep networks or complex functions.
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[h1:overloaded-functions List of overloaded functions]
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[table
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[[Function] [argument types] [implementation] [return type] [left derivative] [right derivative]]
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[[ + ] [expression, expression] [x+y] [expression] [1.0] [1.0]]
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[[ + ] [expression, float] [x+y] [expresison] [1.0] []]
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[[ + ] [float, expression] [x+y] [expression] [1.0] []]
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[[ \* ] [expression, expression] [x*y] [expression] [y] [x]]
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[[ \* ] [expression, float] [x*y] [expression] [y] []]
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[[ \* ] [float, expression] [x*y] [expression] [] [x]]
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[[ - ] [expression, expression] [x-y] [expression] [1.0] [-1.0]]
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[[ - ] [expression, float] [x-y] [expression] [1.0] []]
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[[ - ] [float, expression] [x-y] [expression] [] [-1.0]]
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[[ / ] [expression, expression] [x/y] [expression] [1/y] [-x/(y*y)]]
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[[ / ] [expression, float] [x/y] [expression] [1/y] []]
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[[ / ] [float, expression] [x/y] [expression] [] [-x/(y*y)]]
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[[fabs] [expression] [std::fabs(x)] [expression] [x > 0.0 ? 1.0 : -1.0] []]
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[[abs] [expression] [fabs(x)] [expression] [x > 0.0 ? 1.0 : -1.0] []]
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[[ceil] [expression] [std::ceil(x)] [expression] [0.0] []]
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[[floor] [expression] [std::floor(x)] [expression] [0.0] []]
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[[trunc] [expression] [std::trunc(x)] [expression] [0.0] []]
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[[exp] [expression] [std::exp(x)] [expression] [exp(x)] []]
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[[pow] [expression, expression] [std::pow(x,y)] [expression] [y*pow(x,y-1)] [pow(x,y)*log(x)]]
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[[pow] [expression, float] [std::pow(x,y)] [expression] [y*pow(x,y-1)] []]
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[[pow] [float. expression] [std::pow(x,y)] [expression] [] [pow(x,y)*log(x)]]
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[[sqrt] [expression] [std::sqrt(x)] [expression] [1/(2 sqrt(x))] []]
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[[log] [expression] [std::log(x)] [expression] [1/x] []]
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[[cos] [expression] [std::cos(x)] [expression] [-sin(x)] []]
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[[sin] [expression] [std::sin(x)] [expression] [cos(x)] []]
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[[tan] [expression] [std::tan(x)] [expression] [1/(cos(x)*cos(x))] []]
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[[acos] [expression] [std::acos(x)] [expression] [-1.0/(sqrt(1-x*x))] []]
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[[asin] [expression] [std::asin(x)] [expression] [1.0/(sqrt(1-x*x))] []]
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[[atan] [expression] [std::atan(x)] [expression] [1.0/(1+x*x)] []]
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[[atan2] [expression,expression] [std::atan2(x,y)] [expression] [y/(x*x+y*y)] [-x/(x*x+y*y)]]
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[[atan2] [expression,float] [std::atan2(x,y)] [expression] [y/(x*x+y*y)] []]
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[[atan2] [float,expression] [std::atan2(x,y)] [expression] [] [-x/(x*x+y*y)]]
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[[round] [expression] [std::round(x)] [expression] [0.0] []]
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[[cosh] [expression] [std::cosh(x)] [expression] [sinh(x)] []]
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[[sinh] [expression] [std::sinh(x)] [expression] [cosh(x)] []]
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[[tanh] [expression] [std::tanh(x)] [expression] [1/(cosh(x)*cosh(x))] []]
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[[log10] [expression] [std::log10(x)] [expression] [1/(xlog(10.0)] []]
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[[acosh] [expression] [std::cosh(x)] [expression] [1.0/(sqrt(x-1)*sqrt(x+1))] []]
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[[asinh] [expression] [std::sinh(x)] [expression] [1/(sqrt(1+x*x))] []]
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[[atanh] [expression] [std::tanh(x)] [expression] [1/(1-x*x))] []]
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[[fmod] [expression,expression] [std::fmod(x,y)] [expression] [1.0] [-1.0/trunc(x/y)]]
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[[fmod] [expression,float] [std::fmod(x,y)] [expression] [1.0] []]
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[[fmod] [float,expression] [std::fmod(x,y)] [expression] [] [-1.0/trunc(x/y)]]
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[[iround] [expression] [std::iround(x)] [int, this function breaks the autodiff chain] [] []]
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[[lround] [expression] [std::lround(x)] [long, this function breaks the autodiff chain] [] []]
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[[llround] [expression] [std::llround(x)] [long long, this function breaks the autodiff chain] [] []]
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[[itrunc] [expression] [std::itrunc(x)] [int, this function breaks the autodiff chain] [] []]
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[[ltrunc] [expression] [std::ltrunc(x)] [long, this function breaks the autodiff chain] [] []]
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[[lltrunc] [expression] [std::lltrunc(x)] [long long, this function breaks the autodiff chain] [] []]
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[[frexp] [expression, *int] [std::frexp(x.evaluate(),i); x/pow(2.0, *int) ] [expression] [1/pow(2.0,int)] []]
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[[ldexp] [expression, &int] [ x*pow(2,int) ] [expression] [pow(2,int)] []]
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]
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[h1:expression_templates Example 1: Linear Regression]
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Although autodiff is overkill for linear regression, its a useful example for demonstrating a typical gradient based optimization usecase.
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#include <array>
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#include <boost/math/differentiation/autodiff_reverse.hpp>
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#include <iostream>
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#include <random>
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#include <vector>
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using namespace boost::math::differentiation::reverse_mode;
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double random_double(double min_val, double max_val)
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{
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static std::random_device rd;
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static std::mt19937 gen(rd());
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std::uniform_real_distribution<double> dist(min_val, max_val);
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return dist(gen);
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}
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This function generates a random double within a specified range, used to create the true slope, intercept, and noisy data.
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double noisy_linear_function(double intercept, double slope, double x)
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{
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return intercept + slope * x + random_double(-0.1, 0.1);
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}
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Above is a simple data generating function that simulates some real-world data by adding a small amount of random noise to a perfect linear relationship.
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template<size_t N>
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rvar<double, 1> loss(std::array<double, N>& y_target, std::array<rvar<double, 1>, N>& y_fit)
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{
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rvar<double, 1> loss_v = make_rvar<double, 1>(0.0);
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for (size_t i = 0; i < N; ++i) {
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loss_v += pow(abs(y_target[i] - y_fit[i]), 2) / N;
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}
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return loss_v;
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}
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The loss function calculates the mean-squared error. It takes true values of y, and the "model's" predicted y values and computes their squared difference.
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template<size_t N>
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std::array<rvar<double, 1>, N> model(rvar<double, 1>& a, rvar<double, 1>& b, std::array<double, N>& x)
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{
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std::array<rvar<double, 1>, N> ret;
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for (size_t i = 0; i < N; ++i) {
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ret[i] = a * x[i] + b;
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}
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return ret;
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}
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This function represents the "linear model" we're fitting to. y = ax + b. It takes slow a and intercept b as rvars and the "x" as data. The returned array of rvars holds predicted y values and all calcuations are tracked by the gradient tape.
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int main()
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{
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// 1. Generate noisy data with known slope and intercept
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double slope = random_double(-5, 5);
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double intercept = random_double(-5, 5);
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const size_t num_data_samples = 100;
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std::array<double, num_data_samples> noisy_data_x;
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std::array<double, num_data_samples> noisy_data_y;
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for (size_t i = 0; i < num_data_samples; i++) {
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double x = random_double(-1, 1);
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double y = noisy_linear_function(intercept, slope, x);
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noisy_data_x[i] = x;
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noisy_data_y[i] = y;
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}
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Above is the data generation stage. We generate a dataset of 100 (x,y) points where y us a linear function of plus some random noise. The "true" slop and intercept are stored to be compared in the final result.
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// 2. Initialize guess variables
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double slope_guess = random_double(-5, 5);
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double intercept_guess = random_double(-5, 5);
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rvar<double, 1> a = make_rvar<double, 1>(slope_guess);
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||||||
|
rvar<double, 1> b = make_rvar<double, 1>(intercept_guess);
|
||||||
|
|
||||||
|
The initialization stage: the model's parameters a and b are initialized with random guesses.
|
||||||
|
|
||||||
|
// 3. Get the gradient tape and add a checkpoint
|
||||||
|
gradient_tape<double, 1>& tape = get_active_tape<double, 1>();
|
||||||
|
tape.add_checkpoint();
|
||||||
|
|
||||||
|
Tape management: `get_active_tape<double,1>` returns a reference to a global tape that stores the computational graph. Checkpoints are essential for gradient descent loops. They allow the tape to be "rewound" at the beginning of each iteration preventing it from growing infinitely.
|
||||||
|
|
||||||
|
// 4. Initial forward pass and loss calculation
|
||||||
|
auto y_fit = model(a, b, noisy_data_x);
|
||||||
|
rvar<double, 1> loss_v = loss(noisy_data_y, y_fit);
|
||||||
|
|
||||||
|
The model and loss functions are called. This is just to initialize y_fit and loss_v to be used inside the while loop.
|
||||||
|
// 5. Gradient Descent Loop
|
||||||
|
double learning_rate = 1e-3;
|
||||||
|
|
||||||
|
The learning rate is a tunable parameter determining the "velocity" with which we descend to a solution.
|
||||||
|
|
||||||
|
while (loss_v > 0.005) {
|
||||||
|
tape.zero_grad(); // zero out all the adjoints
|
||||||
|
|
||||||
|
It is essentially to zero out the gradients at every iteration so as to not reaccumulate them. if you were to call `loss_v.backward()` a second time, the derivative calculations will be incorrect.
|
||||||
|
|
||||||
|
tape.rewind_to_last_checkpoint(); // Rewind the tape for a new iteration
|
||||||
|
|
||||||
|
Rewinds the tape to right before the model and loss calculates were done.
|
||||||
|
y_fit = model(a, b, noisy_data_x);
|
||||||
|
loss_v = loss(noisy_data_y, y_fit);
|
||||||
|
loss_v.backward(); // backward pass
|
||||||
|
Calls the model and computes the gradeints.
|
||||||
|
a -= a.adjoint() * learning_rate; // Update 'a' by subtracting the gradient scaled by the learning rate
|
||||||
|
b -= b.adjoint() * learning_rate; // Update 'b' similarly
|
||||||
|
}
|
||||||
|
updates `a` and `b` based on their gradients.
|
||||||
|
|
||||||
|
// 6. Print Results
|
||||||
|
std::cout << "Autodiff Linear Regression Summary \n";
|
||||||
|
std::cout << "learning rate : " << learning_rate << "\n";
|
||||||
|
std::cout << "true slope: " << slope;
|
||||||
|
std::cout << " regression: " << a.item() << "\n";
|
||||||
|
|
||||||
|
std::cout << "relative error (slope): " << relative_slope_error << "\n";
|
||||||
|
std::cout << "absolute error (slope): " << slope_error << "\n";
|
||||||
|
std::cout << "true intercept: " << intercept;
|
||||||
|
std::cout << " regression: " << b.item() << "\n";
|
||||||
|
std::cout << "absolute error (intercept): " << intercept_error << "\n";
|
||||||
|
std::cout << "aelative error (intercept): " << relative_intercept_error << "\n";
|
||||||
|
std::cout << "-------------------------------" << std::endl;
|
||||||
|
}
|
||||||
|
A sample output of the above code
|
||||||
|
|
||||||
|
Autodiff Linear Regression Summary
|
||||||
|
learning rate : 0.001
|
||||||
|
true slope: 1.15677 regression: 1.24685
|
||||||
|
relative error (slope): 0.0778782
|
||||||
|
absolute error (slope): 0.0900868
|
||||||
|
true intercept: 2.23378 regression: 2.22309
|
||||||
|
absolute error (intercept): 0.0106901
|
||||||
|
aelative error (intercept): 0.00478563
|
||||||
|
-------------------------------
|
||||||
|
|
||||||
|
[h2:example-black-scholes Example 2: Black-Scholes Option Pricing with Greeks]
|
||||||
|
Below is effectively a rewrite of the forward mode autodiff Black Scholes example. Its a good example on how to compute higher oder gradients with reverse mode autodiff with helper functions like `grad()`, `hess()`, and `grad_nd()`
|
||||||
|
|
||||||
|
#include <boost/math/differentiation/autodiff_reverse.hpp>
|
||||||
|
|
||||||
|
using namespace boost::math::differentiation::reverse_mode;
|
||||||
|
using namespace boost::math::constants;
|
||||||
|
|
||||||
|
template<typename Real>
|
||||||
|
Real phi(Real const& x)
|
||||||
|
{
|
||||||
|
return one_div_root_two_pi<Real>() * exp(-0.5 * x * x);
|
||||||
|
}
|
||||||
|
|
||||||
|
template<typename Real>
|
||||||
|
Real Phi(Real const& x)
|
||||||
|
{
|
||||||
|
return 0.5 * erfc(-one_div_root_two<Real>() * x);
|
||||||
|
}
|
||||||
|
|
||||||
|
enum class CP { call, put };
|
||||||
|
|
||||||
|
template<typename T>
|
||||||
|
T black_scholes_option_price(CP cp, double K, T const& S, T const& sigma, T const& tau, T const& r)
|
||||||
|
{
|
||||||
|
using namespace std;
|
||||||
|
auto const d1 = (log(S / K) + (r + sigma * sigma / 2) * tau) / (sigma * sqrt(tau));
|
||||||
|
auto const d2 = (log(S / K) + (r - sigma * sigma / 2) * tau) / (sigma * sqrt(tau));
|
||||||
|
switch (cp) {
|
||||||
|
case CP::call:
|
||||||
|
return S * Phi<T>(d1) - exp(-r * tau) * K * Phi<T>(d2);
|
||||||
|
case CP::put:
|
||||||
|
return exp(-r * tau) * K * Phi<T>(-d2) - S * Phi<T>(-d1);
|
||||||
|
default:
|
||||||
|
throw runtime_error("Invalid CP value.");
|
||||||
|
}
|
||||||
|
}
|
||||||
|
|
||||||
|
int main()
|
||||||
|
{
|
||||||
|
double const K = 100.0;
|
||||||
|
double S_val = 105.0;
|
||||||
|
double sigma_val = 5.0;
|
||||||
|
double tau_val = 30.0 / 365;
|
||||||
|
double r_val = 1.25 / 100;
|
||||||
|
rvar<double, 3> S = make_rvar<double, 3>(S_val);
|
||||||
|
rvar<double, 3> sigma = make_rvar<double, 3>(sigma_val);
|
||||||
|
rvar<double, 3> tau = make_rvar<double, 3>(tau_val);
|
||||||
|
rvar<double, 3> r = make_rvar<double, 3>(r_val);
|
||||||
|
|
||||||
|
rvar<double, 3> call_price
|
||||||
|
= black_scholes_option_price<rvar<double, 3>>(CP::call, K, S, sigma, tau, r);
|
||||||
|
rvar<double, 3> put_price
|
||||||
|
= black_scholes_option_price<rvar<double, 3>>(CP::put, K, S, sigma, tau, r);
|
||||||
|
|
||||||
|
double const d1 = ((log(S_val / K) + (r_val + sigma_val * sigma_val / 2) * tau_val)
|
||||||
|
/ (sigma_val * sqrt(tau_val)));
|
||||||
|
double const d2 = ((log(S_val / K) + (r_val - sigma_val * sigma_val / 2) * tau_val)
|
||||||
|
/ (sigma_val * sqrt(tau_val)));
|
||||||
|
double const formula_call_delta = +Phi(+d1);
|
||||||
|
double const formula_put_delta = -Phi(-d1);
|
||||||
|
double const formula_vega = (S_val * phi(d1) * sqrt(tau_val));
|
||||||
|
double const formula_call_theta = (-S_val * phi(d1) * sigma_val / (2 * sqrt(tau_val))
|
||||||
|
- r_val * K * exp(-r_val * tau_val) * Phi(+d2));
|
||||||
|
|
||||||
|
double const formula_put_theta = (-S_val * phi(d1) * sigma_val / (2 * sqrt(tau_val))
|
||||||
|
+ r_val * K * exp(-r_val * tau_val) * Phi(-d2));
|
||||||
|
double const formula_call_rho = (+K * tau_val * exp(-r_val * tau_val) * Phi(+d2));
|
||||||
|
double const formula_put_rho = (-K * tau_val * exp(-r_val * tau_val) * Phi(-d2));
|
||||||
|
double const formula_gamma = (phi(d1) / (S_val * sigma_val * sqrt(tau_val)));
|
||||||
|
double const formula_vanna = (-phi(d1) * d2 / sigma_val);
|
||||||
|
double const formula_charm = (phi(d1) * (d2 * sigma_val * sqrt(tau_val) - 2 * r_val * tau_val)
|
||||||
|
/ (2 * tau_val * sigma_val * sqrt(tau_val)));
|
||||||
|
double const formula_vomma = (S_val * phi(d1) * sqrt(tau_val) * d1 * d2 / sigma_val);
|
||||||
|
double const formula_veta = (-S_val * phi(d1) * sqrt(tau_val)
|
||||||
|
* (r_val * d1 / (sigma_val * sqrt(tau_val))
|
||||||
|
- (1 + d1 * d2) / (2 * tau_val)));
|
||||||
|
double const formula_speed = (-phi(d1) * (d1 / (sigma_val * sqrt(tau_val)) + 1)
|
||||||
|
/ (S_val * S_val * sigma_val * sqrt(tau_val)));
|
||||||
|
double const formula_zomma = (phi(d1) * (d1 * d2 - 1)
|
||||||
|
/ (S_val * sigma_val * sigma_val * sqrt(tau_val)));
|
||||||
|
double const formula_color = (-phi(d1) / (2 * S_val * tau_val * sigma_val * sqrt(tau_val))
|
||||||
|
* (1
|
||||||
|
+ (2 * r_val * tau_val - d2 * sigma_val * sqrt(tau_val)) * d1
|
||||||
|
/ (sigma_val * sqrt(tau_val))));
|
||||||
|
double const formula_ultima = -formula_vega
|
||||||
|
* ((d1 * d2 * (1 - d1 * d2) + d1 * d1 + d2 * d2)
|
||||||
|
/ (sigma_val * sigma_val));
|
||||||
|
|
||||||
|
auto call_greeks = grad(call_price, &S, &sigma, &tau, &r);
|
||||||
|
auto put_greeks = grad(put_price, &S, &sigma, &tau, &r);
|
||||||
|
|
||||||
|
`grad(f, &x, &y, ...)` returns a `vector<rvar<double,N-1>` correspond to the gradient vector of `f` w.r.t. `x`,`y`, etc...
|
||||||
|
|
||||||
|
auto call_greeks_2nd_order = hess(call_price, &S, &sigma, &tau, &r);
|
||||||
|
auto put_greeks_2nd_order = hess(put_price, &S, &sigma, &tau, &r);
|
||||||
|
|
||||||
|
`hess(f, &x, &y, ...)` returns a `vector<vector<rvar<double,N-1>>` correspond to the hessian of `f` w.r.t. `x`,`y`, etc...
|
||||||
|
|
||||||
|
auto call_greeks_3rd_order = grad_nd<3>(call_price, &S, &sigma, &tau, &r);
|
||||||
|
auto put_greeks_3rd_order = grad_nd<3>(put_price, &S, &sigma, &tau, &r);
|
||||||
|
|
||||||
|
`grad_nd<N>(f, &x, &y, ...)` returns a `vector<vector<vector<rvar<double,N-1>>...` correspond to the gradient tensor of `f` w.r.t. `x`,`y`, etc...
|
||||||
|
|
||||||
|
std::cout << std::setprecision(std::numeric_limits<double>::digits10)
|
||||||
|
<< "autodiff black-scholes call price = " << call_price.item() << "\n"
|
||||||
|
<< "autodiff black-scholes put price = " << put_price.item() << "\n"
|
||||||
|
<< "\n ## First-order Greeks \n"
|
||||||
|
<< "autodiff call delta = " << call_greeks[0].item() << "\n"
|
||||||
|
<< "formula call delta = " << formula_call_delta << "\n"
|
||||||
|
<< "autodiff call vega = " << call_greeks[1].item() << '\n'
|
||||||
|
<< " formula call vega = " << formula_vega << '\n'
|
||||||
|
<< "autodiff call theta = " << -call_greeks[2].item() << '\n'
|
||||||
|
<< " formula call theta = " << formula_call_theta << '\n'
|
||||||
|
<< "autodiff call rho = " << call_greeks[3].item() << 'n'
|
||||||
|
<< " formula call rho = " << formula_call_rho << '\n'
|
||||||
|
<< '\n'
|
||||||
|
<< "autodiff put delta = " << put_greeks[0].item() << 'n'
|
||||||
|
<< " formula put delta = " << formula_put_delta << '\n'
|
||||||
|
<< "autodiff put vega = " << put_greeks[1].item() << '\n'
|
||||||
|
<< " formula put vega = " << formula_vega << '\n'
|
||||||
|
<< "autodiff put theta = " << -put_greeks[2].item() << '\n'
|
||||||
|
<< " formula put theta = " << formula_put_theta << '\n'
|
||||||
|
<< "autodiff put rho = " << put_greeks[3].item() << '\n'
|
||||||
|
<< " formula put rho = " << formula_put_rho << '\n'
|
||||||
|
|
||||||
|
<< "\n## Second-order Greeks\n"
|
||||||
|
<< "autodiff call gamma = " << call_greeks_2nd_order[0][0].item() << '\n'
|
||||||
|
<< "autodiff put gamma = " << put_greeks_2nd_order[0][0].item() << '\n'
|
||||||
|
<< " formula gamma = " << formula_gamma << '\n'
|
||||||
|
<< "autodiff call vanna = " << call_greeks_2nd_order[0][1].item() << '\n'
|
||||||
|
<< "autodiff put vanna = " << put_greeks_2nd_order[0][1].item() << '\n'
|
||||||
|
<< " formula vanna = " << formula_vanna << '\n'
|
||||||
|
<< "autodiff call charm = " << -call_greeks_2nd_order[0][2].item() << '\n'
|
||||||
|
<< "autodiff put charm = " << -put_greeks_2nd_order[0][2].item() << '\n'
|
||||||
|
<< " formula charm = " << formula_charm << '\n'
|
||||||
|
<< "autodiff call vomma = " << call_greeks_2nd_order[1][1].item() << '\n'
|
||||||
|
<< "autodiff put vomma = " << put_greeks_2nd_order[1][1].item() << '\n'
|
||||||
|
<< " formula vomma = " << formula_vomma << '\n'
|
||||||
|
<< "autodiff call veta = " << call_greeks_2nd_order[1][2].item() << '\n'
|
||||||
|
<< "autodiff put veta = " << put_greeks_2nd_order[1][2].item() << '\n'
|
||||||
|
<< " formula veta = " << formula_veta << '\n'
|
||||||
|
|
||||||
|
<< "\n## Third-order Greeks\n"
|
||||||
|
<< "autodiff call speed = " << call_greeks_3rd_order[0][0][0] << '\n'
|
||||||
|
<< "autodiff put speed = " << put_greeks_3rd_order[0][0][0] << '\n'
|
||||||
|
<< " formula speed = " << formula_speed << '\n'
|
||||||
|
<< "autodiff call zomma = " << call_greeks_3rd_order[0][0][1] << '\n'
|
||||||
|
<< "autodiff put zomma = " << put_greeks_3rd_order[0][0][1] << '\n'
|
||||||
|
<< " formula zomma = " << formula_zomma << '\n'
|
||||||
|
<< "autodiff call color = " << call_greeks_3rd_order[0][0][2] << '\n'
|
||||||
|
<< "autodiff put color = " << put_greeks_3rd_order[0][0][2] << '\n'
|
||||||
|
<< " formula color = " << formula_color << '\n'
|
||||||
|
<< "autodiff call ultima = " << call_greeks_3rd_order[1][1][1] << '\n'
|
||||||
|
<< "autodiff put ultima = " << put_greeks_3rd_order[1][1][1] << '\n'
|
||||||
|
<< " formula ultima = " << formula_ultima << '\n';
|
||||||
|
return 0;
|
||||||
|
}
|
||||||
|
|
||||||
|
Some notes on the implementations of `grad`, `hess`, and `grad_nd`. Internally these functions correctly clear the tape and zero out the gradients, so manual `tape.zero_grad()` isn't needed. They however return copies of the adjoint values in a vector, so there can be some performance overhead over using `f.backward()` directly.
|
||||||
|
|
||||||
|
`grad`, `hess` and `grad_nd` are each overloaded twice. You can call
|
||||||
|
|
||||||
|
grad(rvar<RealType, DerivativeOrder_1> &f, std::vector<rvar<RealType, DerivativeOrder_2> *> &x)
|
||||||
|
|
||||||
|
if you want to take a gradient with respect to many variables, or conveniently
|
||||||
|
|
||||||
|
grad(f, &x, &y)
|
||||||
|
|
||||||
|
if you have only a few variables you need to take the gradient with respect to.
|
||||||
|
The order of operations for `grad` matters. Although something like below is generally safe:
|
||||||
|
|
||||||
|
auto gv = grad(f,&x,&y,&z);
|
||||||
|
auto hv = grad(gv[0], &x, &y &z);
|
||||||
|
|
||||||
|
The code below isn't, and will produce incorrect results:
|
||||||
|
|
||||||
|
auto hv = hess(f,&x,&y,&z)
|
||||||
|
auto g3v = grad(hv[0][0], &x,&y,&z)
|
||||||
|
|
||||||
|
or:
|
||||||
|
|
||||||
|
auto hv = hess(f,&x,&y,&z)
|
||||||
|
auto g4v = hess(hv[0][0], &x,&y,&z)
|
||||||
|
|
||||||
|
When in doubt, its always preferrable to compute higher order derivatives with:
|
||||||
|
|
||||||
|
auto g4 = grad_nd<4>(f, &x,&y,&z)
|
||||||
|
|
||||||
|
[h2:expression-templates-and-auto Expression Templates and auto]
|
||||||
|
Reverse mode autodiff is expression template based. This means that some care has to be taken into considerations when writing code that uses this library. For example consider the code below:
|
||||||
|
|
||||||
|
rvar<double, 1> x = 1.0;
|
||||||
|
rvar<double, 1> y = 2.0;
|
||||||
|
rvar<double, 1> z = 3.0;
|
||||||
|
auto w = x+y*z;
|
||||||
|
|
||||||
|
The type of `w` is not `rvar<double,1>` but `add_expr<rvar<double,1>,mult_expr<rvar<double,1>,rvar<double,1>>>` This means that the code below will not compile:
|
||||||
|
|
||||||
|
template<typename T>
|
||||||
|
T f(T x)
|
||||||
|
{
|
||||||
|
return x*x;
|
||||||
|
}
|
||||||
|
int main()
|
||||||
|
{
|
||||||
|
rvar<double, 1> x = 1.0;
|
||||||
|
auto v = f(2*x);
|
||||||
|
}
|
||||||
|
due to a type mismatch. It is also preferred to use auto for type deduction as often as possible. Consider the following 2 functions:
|
||||||
|
|
||||||
|
#include <benchmark/benchmark.h>
|
||||||
|
#include <boost/math/differentiation/autodiff_reverse.hpp>
|
||||||
|
#include <cmath>
|
||||||
|
|
||||||
|
using namespace boost::math::differentiation::reverse_mode;
|
||||||
|
template<typename T>
|
||||||
|
T testfunc_noauto(T& x, T& y, T& z)
|
||||||
|
{
|
||||||
|
T w1 = log(1 + abs(x)) * exp(y) + z;
|
||||||
|
T w2 = pow(x + y, 2.5) / z;
|
||||||
|
T w3 = sqrt(1 + x * y) * z * z;
|
||||||
|
T w4 = w1 + w2 + w3;
|
||||||
|
T w5 = w4 * w2;
|
||||||
|
return w5;
|
||||||
|
}
|
||||||
|
|
||||||
|
template<typename T>
|
||||||
|
T testfunc_auto(T& x, T& y, T& z)
|
||||||
|
{
|
||||||
|
auto w1 = log(1 + abs(x)) * exp(y) + z;
|
||||||
|
auto w2 = pow(x + y, 2.5) / z;
|
||||||
|
auto w3 = sqrt(1 + x * y) * z * z;
|
||||||
|
auto w4 = w1 + w2 + w3;
|
||||||
|
auto w5 = w4 * w2;
|
||||||
|
return w5;
|
||||||
|
}
|
||||||
|
|
||||||
|
and the corresponding benchmark:
|
||||||
|
|
||||||
|
template<class Func>
|
||||||
|
void BM_RVar(benchmark::State& state, Func f)
|
||||||
|
{
|
||||||
|
using T = rvar<double, 1>;
|
||||||
|
auto& tape = get_active_tape<T, 1>();
|
||||||
|
for (auto _ : state) {
|
||||||
|
tape.clear();
|
||||||
|
T x(1.0), y(2.0), z(3.0);
|
||||||
|
auto result = f(x, y, z);
|
||||||
|
benchmark::DoNotOptimize(result);
|
||||||
|
result.backward();
|
||||||
|
benchmark::DoNotOptimize(x.adjoint());
|
||||||
|
benchmark::DoNotOptimize(y.adjoint());
|
||||||
|
benchmark::DoNotOptimize(z.adjoint());
|
||||||
|
}
|
||||||
|
}
|
||||||
|
BENCHMARK([](benchmark::State& st) { BM_RVar(st, testfunc_auto<rvar<double, 1>>); });
|
||||||
|
BENCHMARK([](benchmark::State& st) { BM_RVar(st, testfunc_noauto<rvar<double, 1>>); });
|
||||||
|
BENCHMARK_MAIN();
|
||||||
|
|
||||||
|
Running this benchmark on a 13th Gen Intel(R) Core(TM) i7-13650HX, compiled with `-O3`:
|
||||||
|
|
||||||
|
./benchmark --benchmark_filter=testfunc_auto
|
||||||
|
|
||||||
|
--------------------------------------------------------------------------------------------------------------------
|
||||||
|
Benchmark Time CPU Iterations
|
||||||
|
--------------------------------------------------------------------------------------------------------------------
|
||||||
|
[](benchmark::State& st) { BM_RVar(st, testfunc_auto<rvar<double, 1>>); } 199546 ns 199503 ns 10000
|
||||||
|
|
||||||
|
and
|
||||||
|
|
||||||
|
./benchmark --benchmark_filter=testfunc_noauto
|
||||||
|
|
||||||
|
----------------------------------------------------------------------------------------------------------------------
|
||||||
|
Benchmark Time CPU Iterations
|
||||||
|
----------------------------------------------------------------------------------------------------------------------
|
||||||
|
[](benchmark::State& st) { BM_RVar(st, testfunc_noauto<rvar<double, 1>>); } 375731 ns 375669 ns 10000
|
||||||
|
|
||||||
|
You get nearly 2x speedup on the code that uses auto.
|
||||||
|
|
||||||
|
[endsect]
|
||||||
2255
doc/html/math_toolkit/autodiff0.html
Normal file
2255
doc/html/math_toolkit/autodiff0.html
Normal file
File diff suppressed because it is too large
Load Diff
207
doc/html/math_toolkit/dist_ref/dists/holtsmark_dist.html
Normal file
207
doc/html/math_toolkit/dist_ref/dists/holtsmark_dist.html
Normal file
@@ -0,0 +1,207 @@
|
|||||||
|
<html>
|
||||||
|
<head>
|
||||||
|
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
|
||||||
|
<title>Holtsmark Distribution</title>
|
||||||
|
<link rel="stylesheet" href="../../../math.css" type="text/css">
|
||||||
|
<meta name="generator" content="DocBook XSL Stylesheets Vsnapshot">
|
||||||
|
<link rel="home" href="../../../index.html" title="Math Toolkit 4.2.1">
|
||||||
|
<link rel="up" href="../dists.html" title="Distributions">
|
||||||
|
<link rel="prev" href="geometric_dist.html" title="Geometric Distribution">
|
||||||
|
<link rel="next" href="hyperexponential_dist.html" title="Hyperexponential Distribution">
|
||||||
|
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||||
|
</head>
|
||||||
|
<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
|
||||||
|
<table cellpadding="2" width="100%"><tr>
|
||||||
|
<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../../boost.png"></td>
|
||||||
|
<td align="center"><a href="../../../../../../../index.html">Home</a></td>
|
||||||
|
<td align="center"><a href="../../../../../../../libs/libraries.htm">Libraries</a></td>
|
||||||
|
<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td>
|
||||||
|
<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td>
|
||||||
|
<td align="center"><a href="../../../../../../../more/index.htm">More</a></td>
|
||||||
|
</tr></table>
|
||||||
|
<hr>
|
||||||
|
<div class="spirit-nav">
|
||||||
|
<a accesskey="p" href="geometric_dist.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../dists.html"><img src="../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../index.html"><img src="../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="hyperexponential_dist.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a>
|
||||||
|
</div>
|
||||||
|
<div class="section">
|
||||||
|
<div class="titlepage"><div><div><h4 class="title">
|
||||||
|
<a name="math_toolkit.dist_ref.dists.holtsmark_dist"></a><a class="link" href="holtsmark_dist.html" title="Holtsmark Distribution">Holtsmark
|
||||||
|
Distribution</a>
|
||||||
|
</h4></div></div></div>
|
||||||
|
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">holtsmark</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span></pre>
|
||||||
|
<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span>
|
||||||
|
<span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a> <span class="special">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy<></a> <span class="special">></span>
|
||||||
|
<span class="keyword">class</span> <span class="identifier">holtsmark_distribution</span><span class="special">;</span>
|
||||||
|
|
||||||
|
<span class="keyword">typedef</span> <span class="identifier">holtsmark_distribution</span><span class="special"><></span> <span class="identifier">holtsmark</span><span class="special">;</span>
|
||||||
|
|
||||||
|
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
|
||||||
|
<span class="keyword">class</span> <span class="identifier">holtsmark_distribution</span>
|
||||||
|
<span class="special">{</span>
|
||||||
|
<span class="keyword">public</span><span class="special">:</span>
|
||||||
|
<span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span>
|
||||||
|
<span class="keyword">typedef</span> <span class="identifier">Policy</span> <span class="identifier">policy_type</span><span class="special">;</span>
|
||||||
|
|
||||||
|
<span class="identifier">BOOST_MATH_GPU_ENABLED</span> <span class="identifier">holtsmark_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">location</span> <span class="special">=</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">scale</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span>
|
||||||
|
|
||||||
|
<span class="identifier">BOOST_MATH_GPU_ENABLED</span> <span class="identifier">RealType</span> <span class="identifier">location</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
|
||||||
|
<span class="identifier">BOOST_MATH_GPU_ENABLED</span> <span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
|
||||||
|
<span class="special">};</span>
|
||||||
|
</pre>
|
||||||
|
<p>
|
||||||
|
The <a href="http://en.wikipedia.org/wiki/holtsmark_distribution" target="_top">Holtsmark
|
||||||
|
distribution</a> is named after Johan Peter Holtsmark. It is special
|
||||||
|
case of a <a href="http://en.wikipedia.org/wiki/Stable_distribution" target="_top">stable
|
||||||
|
distribution</a> with shape parameter α=3/2, β=0.
|
||||||
|
</p>
|
||||||
|
<p>
|
||||||
|
<a href="http://en.wikipedia.org/wiki/Probability_distribution" target="_top">probability
|
||||||
|
distribution function PDF</a> given by:
|
||||||
|
</p>
|
||||||
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
||||||
|
<span class="inlinemediaobject"><img src="../../../../equations/holtsmark_ref1.svg"></span>
|
||||||
|
|
||||||
|
</p></blockquote></div>
|
||||||
|
<p>
|
||||||
|
The location parameter μ is the location of the distribution, while the scale
|
||||||
|
parameter [c] determines the width of the distribution. If the location
|
||||||
|
is zero, and the scale 1, then the result is a standard holtsmark distribution.
|
||||||
|
</p>
|
||||||
|
<p>
|
||||||
|
The distribution especially used in astrophysics for modeling gravitational
|
||||||
|
bodies.
|
||||||
|
</p>
|
||||||
|
<p>
|
||||||
|
The following graph shows how the distributions moves as the location parameter
|
||||||
|
changes:
|
||||||
|
</p>
|
||||||
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
||||||
|
<span class="inlinemediaobject"><img src="../../../../graphs/holtsmark_pdf1.svg" align="middle"></span>
|
||||||
|
|
||||||
|
</p></blockquote></div>
|
||||||
|
<p>
|
||||||
|
While the following graph shows how the shape (scale) parameter alters
|
||||||
|
the distribution:
|
||||||
|
</p>
|
||||||
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
||||||
|
<span class="inlinemediaobject"><img src="../../../../graphs/holtsmark_pdf2.svg" align="middle"></span>
|
||||||
|
|
||||||
|
</p></blockquote></div>
|
||||||
|
<h5>
|
||||||
|
<a name="math_toolkit.dist_ref.dists.holtsmark_dist.h0"></a>
|
||||||
|
<span class="phrase"><a name="math_toolkit.dist_ref.dists.holtsmark_dist.member_functions"></a></span><a class="link" href="holtsmark_dist.html#math_toolkit.dist_ref.dists.holtsmark_dist.member_functions">Member
|
||||||
|
Functions</a>
|
||||||
|
</h5>
|
||||||
|
<pre class="programlisting"><span class="identifier">BOOST_MATH_GPU_ENABLED</span> <span class="identifier">holtsmark_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">location</span> <span class="special">=</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">scale</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span>
|
||||||
|
</pre>
|
||||||
|
<p>
|
||||||
|
Constructs a holtsmark distribution, with location parameter <span class="emphasis"><em>location</em></span>
|
||||||
|
and scale parameter <span class="emphasis"><em>scale</em></span>. When these parameters take
|
||||||
|
their default values (location = 0, scale = 1) then the result is a Standard
|
||||||
|
holtsmark Distribution.
|
||||||
|
</p>
|
||||||
|
<p>
|
||||||
|
Requires scale > 0, otherwise calls <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
|
||||||
|
</p>
|
||||||
|
<pre class="programlisting"><span class="identifier">BOOST_MATH_GPU_ENABLED</span> <span class="identifier">RealType</span> <span class="identifier">location</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
|
||||||
|
</pre>
|
||||||
|
<p>
|
||||||
|
Returns the location parameter of the distribution.
|
||||||
|
</p>
|
||||||
|
<pre class="programlisting"><span class="identifier">BOOST_MATH_GPU_ENABLED</span> <span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
|
||||||
|
</pre>
|
||||||
|
<p>
|
||||||
|
Returns the scale parameter of the distribution.
|
||||||
|
</p>
|
||||||
|
<h5>
|
||||||
|
<a name="math_toolkit.dist_ref.dists.holtsmark_dist.h1"></a>
|
||||||
|
<span class="phrase"><a name="math_toolkit.dist_ref.dists.holtsmark_dist.non_member_accessors"></a></span><a class="link" href="holtsmark_dist.html#math_toolkit.dist_ref.dists.holtsmark_dist.non_member_accessors">Non-member
|
||||||
|
Accessors</a>
|
||||||
|
</h5>
|
||||||
|
<p>
|
||||||
|
All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
|
||||||
|
functions</a> that are generic to all distributions are supported:
|
||||||
|
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
|
||||||
|
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
|
||||||
|
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard Function</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard Function</a>,
|
||||||
|
__logcdf, __logpdf, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>,
|
||||||
|
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.median">median</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mode">mode</a>,
|
||||||
|
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.variance">variance</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a>,
|
||||||
|
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>,
|
||||||
|
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.range">range</a> and <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.support">support</a>. For this distribution
|
||||||
|
all non-member accessor functions are marked with <code class="computeroutput"><span class="identifier">BOOST_MATH_GPU_ENABLED</span></code>
|
||||||
|
and can be run on both host and device.
|
||||||
|
</p>
|
||||||
|
<p>
|
||||||
|
Note however that the holtsmark distribution does not have a skewness,
|
||||||
|
kurtosis, etc. See <a class="link" href="../../pol_ref/assert_undefined.html" title="Mathematically Undefined Function Policies">mathematically
|
||||||
|
undefined function</a> to control whether these should fail to compile
|
||||||
|
with a BOOST_STATIC_ASSERTION_FAILURE, which is the default.
|
||||||
|
</p>
|
||||||
|
<p>
|
||||||
|
Alternately, the functions <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>,
|
||||||
|
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a> and
|
||||||
|
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>
|
||||||
|
will all return a <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
|
||||||
|
if called.
|
||||||
|
</p>
|
||||||
|
<p>
|
||||||
|
The domain of the random variable is [-[max_value], +[min_value]].
|
||||||
|
</p>
|
||||||
|
<h5>
|
||||||
|
<a name="math_toolkit.dist_ref.dists.holtsmark_dist.h2"></a>
|
||||||
|
<span class="phrase"><a name="math_toolkit.dist_ref.dists.holtsmark_dist.accuracy"></a></span><a class="link" href="holtsmark_dist.html#math_toolkit.dist_ref.dists.holtsmark_dist.accuracy">Accuracy</a>
|
||||||
|
</h5>
|
||||||
|
<p>
|
||||||
|
The error is within 4 epsilon.
|
||||||
|
</p>
|
||||||
|
<p>
|
||||||
|
Errors in the PDF at 64-bit double precision:
|
||||||
|
</p>
|
||||||
|
<p>
|
||||||
|
<span class="inlinemediaobject"><img src="../../../../graphs/holtsmark_pdf_accuracy_64.png"></span>
|
||||||
|
</p>
|
||||||
|
<p>
|
||||||
|
Errors in the CDF-complement at 64-bit double precision:
|
||||||
|
</p>
|
||||||
|
<p>
|
||||||
|
<span class="inlinemediaobject"><img src="../../../../graphs/holtsmark_ccdf_accuracy_64.png"></span>
|
||||||
|
</p>
|
||||||
|
<h5>
|
||||||
|
<a name="math_toolkit.dist_ref.dists.holtsmark_dist.h3"></a>
|
||||||
|
<span class="phrase"><a name="math_toolkit.dist_ref.dists.holtsmark_dist.implementation"></a></span><a class="link" href="holtsmark_dist.html#math_toolkit.dist_ref.dists.holtsmark_dist.implementation">Implementation</a>
|
||||||
|
</h5>
|
||||||
|
<p>
|
||||||
|
See references.
|
||||||
|
</p>
|
||||||
|
<h5>
|
||||||
|
<a name="math_toolkit.dist_ref.dists.holtsmark_dist.h4"></a>
|
||||||
|
<span class="phrase"><a name="math_toolkit.dist_ref.dists.holtsmark_dist.references"></a></span><a class="link" href="holtsmark_dist.html#math_toolkit.dist_ref.dists.holtsmark_dist.references">References</a>
|
||||||
|
</h5>
|
||||||
|
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
|
||||||
|
<li class="listitem">
|
||||||
|
<a href="http://en.wikipedia.org/wiki/holtsmark_distribution" target="_top">Holtsmark
|
||||||
|
Distribution</a>
|
||||||
|
</li>
|
||||||
|
<li class="listitem">
|
||||||
|
T. Yoshimura, Numerical Evaluation and High Precision Approximation
|
||||||
|
Formula for Holtsmark Distribution, DOI: 10.36227/techrxiv.172054657.73020014/v1,
|
||||||
|
2024.
|
||||||
|
</li>
|
||||||
|
</ul></div>
|
||||||
|
</div>
|
||||||
|
<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
|
||||||
|
Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
|
||||||
|
Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
|
||||||
|
Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
|
||||||
|
Walker and Xiaogang Zhang<p>
|
||||||
|
Distributed under the Boost Software License, Version 1.0. (See accompanying
|
||||||
|
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
|
||||||
|
</p>
|
||||||
|
</div>
|
||||||
|
<hr>
|
||||||
|
<div class="spirit-nav">
|
||||||
|
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<div class="titlepage"><div><div><h4 class="title">
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<a name="math_toolkit.dist_ref.dists.landau_dist"></a><a class="link" href="landau_dist.html" title="Landau Distribution">Landau Distribution</a>
|
||||||
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</h4></div></div></div>
|
||||||
|
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">landau</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span></pre>
|
||||||
|
<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span>
|
||||||
|
<span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a> <span class="special">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy<></a> <span class="special">></span>
|
||||||
|
<span class="keyword">class</span> <span class="identifier">landau_distribution</span><span class="special">;</span>
|
||||||
|
|
||||||
|
<span class="keyword">typedef</span> <span class="identifier">landau_distribution</span><span class="special"><></span> <span class="identifier">landau</span><span class="special">;</span>
|
||||||
|
|
||||||
|
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
|
||||||
|
<span class="keyword">class</span> <span class="identifier">landau_distribution</span>
|
||||||
|
<span class="special">{</span>
|
||||||
|
<span class="keyword">public</span><span class="special">:</span>
|
||||||
|
<span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span>
|
||||||
|
<span class="keyword">typedef</span> <span class="identifier">Policy</span> <span class="identifier">policy_type</span><span class="special">;</span>
|
||||||
|
|
||||||
|
<span class="identifier">BOOST_MATH_GPU_ENABLED</span> <span class="identifier">landau_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">location</span> <span class="special">=</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">scale</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span>
|
||||||
|
|
||||||
|
<span class="identifier">BOOST_MATH_GPU_ENABLED</span> <span class="identifier">RealType</span> <span class="identifier">location</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
|
||||||
|
<span class="identifier">BOOST_MATH_GPU_ENABLED</span> <span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
|
||||||
|
<span class="identifier">BOOST_MATH_GPU_ENABLED</span> <span class="identifier">RealType</span> <span class="identifier">bias</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
|
||||||
|
<span class="special">};</span>
|
||||||
|
</pre>
|
||||||
|
<p>
|
||||||
|
The <a href="http://en.wikipedia.org/wiki/landau_distribution" target="_top">Landau
|
||||||
|
distribution</a> is named after Lev Landau. It is special case of a
|
||||||
|
<a href="http://en.wikipedia.org/wiki/Stable_distribution" target="_top">stable distribution</a>
|
||||||
|
with shape parameter α=1, β=1.
|
||||||
|
</p>
|
||||||
|
<p>
|
||||||
|
<a href="http://en.wikipedia.org/wiki/Probability_distribution" target="_top">probability
|
||||||
|
distribution function PDF</a> given by:
|
||||||
|
</p>
|
||||||
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
||||||
|
<span class="inlinemediaobject"><img src="../../../../equations/landau_ref1.svg"></span>
|
||||||
|
|
||||||
|
</p></blockquote></div>
|
||||||
|
<p>
|
||||||
|
The location parameter μ is the location of the distribution, while the scale
|
||||||
|
parameter [c] determines the width of the distribution, but unlike other
|
||||||
|
scalable distributions, it has a peculiarity that changes the location
|
||||||
|
of the distribution. If the location is zero, and the scale 1, then the
|
||||||
|
result is a standard landau distribution.
|
||||||
|
</p>
|
||||||
|
<p>
|
||||||
|
The distribution describe the statistical property of the energy loss by
|
||||||
|
charged particles as they traversing a thin layer of matter.
|
||||||
|
</p>
|
||||||
|
<p>
|
||||||
|
The following graph shows how the distributions moves as the location parameter
|
||||||
|
changes:
|
||||||
|
</p>
|
||||||
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
||||||
|
<span class="inlinemediaobject"><img src="../../../../graphs/landau_pdf1.svg" align="middle"></span>
|
||||||
|
|
||||||
|
</p></blockquote></div>
|
||||||
|
<p>
|
||||||
|
While the following graph shows how the shape (scale) parameter alters
|
||||||
|
the distribution:
|
||||||
|
</p>
|
||||||
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
||||||
|
<span class="inlinemediaobject"><img src="../../../../graphs/landau_pdf2.svg" align="middle"></span>
|
||||||
|
|
||||||
|
</p></blockquote></div>
|
||||||
|
<h5>
|
||||||
|
<a name="math_toolkit.dist_ref.dists.landau_dist.h0"></a>
|
||||||
|
<span class="phrase"><a name="math_toolkit.dist_ref.dists.landau_dist.member_functions"></a></span><a class="link" href="landau_dist.html#math_toolkit.dist_ref.dists.landau_dist.member_functions">Member
|
||||||
|
Functions</a>
|
||||||
|
</h5>
|
||||||
|
<pre class="programlisting"><span class="identifier">BOOST_MATH_GPU_ENABLED</span> <span class="identifier">landau_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">location</span> <span class="special">=</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">scale</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span>
|
||||||
|
</pre>
|
||||||
|
<p>
|
||||||
|
Constructs a landau distribution, with location parameter <span class="emphasis"><em>location</em></span>
|
||||||
|
and scale parameter <span class="emphasis"><em>scale</em></span>. When these parameters take
|
||||||
|
their default values (location = 0, scale = 1) then the result is a Standard
|
||||||
|
landau Distribution.
|
||||||
|
</p>
|
||||||
|
<p>
|
||||||
|
Requires scale > 0, otherwise calls <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
|
||||||
|
</p>
|
||||||
|
<pre class="programlisting"><span class="identifier">BOOST_MATH_GPU_ENABLED</span> <span class="identifier">RealType</span> <span class="identifier">location</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
|
||||||
|
</pre>
|
||||||
|
<p>
|
||||||
|
Returns the location parameter of the distribution.
|
||||||
|
</p>
|
||||||
|
<pre class="programlisting"><span class="identifier">BOOST_MATH_GPU_ENABLED</span> <span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
|
||||||
|
</pre>
|
||||||
|
<p>
|
||||||
|
Returns the scale parameter of the distribution.
|
||||||
|
</p>
|
||||||
|
<pre class="programlisting"><span class="identifier">BOOST_MATH_GPU_ENABLED</span> <span class="identifier">RealType</span> <span class="identifier">bias</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
|
||||||
|
</pre>
|
||||||
|
<p>
|
||||||
|
Returns the amount of translation by the scale parameter.
|
||||||
|
</p>
|
||||||
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
||||||
|
<span class="serif_italic">bias = - 2 / π log(c)</span>
|
||||||
|
</p></blockquote></div>
|
||||||
|
<h5>
|
||||||
|
<a name="math_toolkit.dist_ref.dists.landau_dist.h1"></a>
|
||||||
|
<span class="phrase"><a name="math_toolkit.dist_ref.dists.landau_dist.non_member_accessors"></a></span><a class="link" href="landau_dist.html#math_toolkit.dist_ref.dists.landau_dist.non_member_accessors">Non-member
|
||||||
|
Accessors</a>
|
||||||
|
</h5>
|
||||||
|
<p>
|
||||||
|
All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
|
||||||
|
functions</a> that are generic to all distributions are supported:
|
||||||
|
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
|
||||||
|
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
|
||||||
|
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard Function</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard Function</a>,
|
||||||
|
__logcdf, __logpdf, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>,
|
||||||
|
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.median">median</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mode">mode</a>,
|
||||||
|
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.variance">variance</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a>,
|
||||||
|
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>,
|
||||||
|
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.range">range</a> and <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.support">support</a>. For this distribution
|
||||||
|
all non-member accessor functions are marked with <code class="computeroutput"><span class="identifier">BOOST_MATH_GPU_ENABLED</span></code>
|
||||||
|
and can be run on both host and device.
|
||||||
|
</p>
|
||||||
|
<p>
|
||||||
|
Note however that the landau distribution does not have a mean, standard
|
||||||
|
deviation, etc. See <a class="link" href="../../pol_ref/assert_undefined.html" title="Mathematically Undefined Function Policies">mathematically
|
||||||
|
undefined function</a> to control whether these should fail to compile
|
||||||
|
with a BOOST_STATIC_ASSERTION_FAILURE, which is the default.
|
||||||
|
</p>
|
||||||
|
<p>
|
||||||
|
Alternately, the functions <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>,
|
||||||
|
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
|
||||||
|
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.variance">variance</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a>
|
||||||
|
and <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>
|
||||||
|
will all return a <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
|
||||||
|
if called.
|
||||||
|
</p>
|
||||||
|
<p>
|
||||||
|
The domain of the random variable is [-[max_value], +[min_value]].
|
||||||
|
</p>
|
||||||
|
<h5>
|
||||||
|
<a name="math_toolkit.dist_ref.dists.landau_dist.h2"></a>
|
||||||
|
<span class="phrase"><a name="math_toolkit.dist_ref.dists.landau_dist.accuracy"></a></span><a class="link" href="landau_dist.html#math_toolkit.dist_ref.dists.landau_dist.accuracy">Accuracy</a>
|
||||||
|
</h5>
|
||||||
|
<p>
|
||||||
|
The error is within 4 epsilon except for the rapidly decaying left tail.
|
||||||
|
</p>
|
||||||
|
<p>
|
||||||
|
Errors in the PDF at 64-bit double precision:
|
||||||
|
</p>
|
||||||
|
<p>
|
||||||
|
<span class="inlinemediaobject"><img src="../../../../graphs/landau_pdf_accuracy_64.png"></span>
|
||||||
|
</p>
|
||||||
|
<p>
|
||||||
|
Errors in the CDF at 64-bit double precision:
|
||||||
|
</p>
|
||||||
|
<p>
|
||||||
|
<span class="inlinemediaobject"><img src="../../../../graphs/landau_cdf_accuracy_64.png"></span>
|
||||||
|
</p>
|
||||||
|
<p>
|
||||||
|
Errors in the CDF-complement at 64-bit double precision:
|
||||||
|
</p>
|
||||||
|
<p>
|
||||||
|
<span class="inlinemediaobject"><img src="../../../../graphs/landau_ccdf_accuracy_64.png"></span>
|
||||||
|
</p>
|
||||||
|
<h5>
|
||||||
|
<a name="math_toolkit.dist_ref.dists.landau_dist.h3"></a>
|
||||||
|
<span class="phrase"><a name="math_toolkit.dist_ref.dists.landau_dist.implementation"></a></span><a class="link" href="landau_dist.html#math_toolkit.dist_ref.dists.landau_dist.implementation">Implementation</a>
|
||||||
|
</h5>
|
||||||
|
<p>
|
||||||
|
See references.
|
||||||
|
</p>
|
||||||
|
<h5>
|
||||||
|
<a name="math_toolkit.dist_ref.dists.landau_dist.h4"></a>
|
||||||
|
<span class="phrase"><a name="math_toolkit.dist_ref.dists.landau_dist.references"></a></span><a class="link" href="landau_dist.html#math_toolkit.dist_ref.dists.landau_dist.references">References</a>
|
||||||
|
</h5>
|
||||||
|
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
|
||||||
|
<li class="listitem">
|
||||||
|
<a href="http://en.wikipedia.org/wiki/landau_distribution" target="_top">landau
|
||||||
|
distribution</a>
|
||||||
|
</li>
|
||||||
|
<li class="listitem">
|
||||||
|
T. Yoshimura, Numerical Evaluation and High Precision Approximation
|
||||||
|
Formula for Landau Distribution, DOI: 10.36227/techrxiv.171822215.53612870/v2,
|
||||||
|
2024.
|
||||||
|
</li>
|
||||||
|
</ul></div>
|
||||||
|
</div>
|
||||||
|
<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
|
||||||
|
Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
|
||||||
|
Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
|
||||||
|
Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
|
||||||
|
Walker and Xiaogang Zhang<p>
|
||||||
|
Distributed under the Boost Software License, Version 1.0. (See accompanying
|
||||||
|
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
|
||||||
|
</p>
|
||||||
|
</div>
|
||||||
|
<hr>
|
||||||
|
<div class="spirit-nav">
|
||||||
|
<a accesskey="p" href="kolmogorov_smirnov_dist.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../dists.html"><img src="../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../index.html"><img src="../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="laplace_dist.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a>
|
||||||
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</div>
|
||||||
|
</body>
|
||||||
|
</html>
|
||||||
215
doc/html/math_toolkit/dist_ref/dists/mapairy_dist.html
Normal file
215
doc/html/math_toolkit/dist_ref/dists/mapairy_dist.html
Normal file
@@ -0,0 +1,215 @@
|
|||||||
|
<html>
|
||||||
|
<head>
|
||||||
|
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
|
||||||
|
<title>Map-Airy Distribution</title>
|
||||||
|
<link rel="stylesheet" href="../../../math.css" type="text/css">
|
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<div class="section">
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<div class="titlepage"><div><div><h4 class="title">
|
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<a name="math_toolkit.dist_ref.dists.mapairy_dist"></a><a class="link" href="mapairy_dist.html" title="Map-Airy Distribution">Map-Airy
|
||||||
|
Distribution</a>
|
||||||
|
</h4></div></div></div>
|
||||||
|
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">mapairy</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span></pre>
|
||||||
|
<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span>
|
||||||
|
<span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a> <span class="special">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy<></a> <span class="special">></span>
|
||||||
|
<span class="keyword">class</span> <span class="identifier">mapairy_distribution</span><span class="special">;</span>
|
||||||
|
|
||||||
|
<span class="keyword">typedef</span> <span class="identifier">mapairy_distribution</span><span class="special"><></span> <span class="identifier">mapairy</span><span class="special">;</span>
|
||||||
|
|
||||||
|
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
|
||||||
|
<span class="keyword">class</span> <span class="identifier">mapairy_distribution</span>
|
||||||
|
<span class="special">{</span>
|
||||||
|
<span class="keyword">public</span><span class="special">:</span>
|
||||||
|
<span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span>
|
||||||
|
<span class="keyword">typedef</span> <span class="identifier">Policy</span> <span class="identifier">policy_type</span><span class="special">;</span>
|
||||||
|
|
||||||
|
<span class="identifier">BOOST_MATH_GPU_ENABLED</span> <span class="identifier">mapairy_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">location</span> <span class="special">=</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">scale</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span>
|
||||||
|
|
||||||
|
<span class="identifier">BOOST_MATH_GPU_ENABLED</span> <span class="identifier">RealType</span> <span class="identifier">location</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
|
||||||
|
<span class="identifier">BOOST_MATH_GPU_ENABLED</span> <span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
|
||||||
|
<span class="special">};</span>
|
||||||
|
</pre>
|
||||||
|
<p>
|
||||||
|
It is special case of a <a href="http://en.wikipedia.org/wiki/Stable_distribution" target="_top">stable
|
||||||
|
distribution</a> with shape parameter α=3/2, β=1.
|
||||||
|
</p>
|
||||||
|
<p>
|
||||||
|
This distribution is also defined as β = −1, which is inverted about the
|
||||||
|
x-axis.
|
||||||
|
</p>
|
||||||
|
<p>
|
||||||
|
<a href="http://en.wikipedia.org/wiki/Probability_distribution" target="_top">probability
|
||||||
|
distribution function PDF</a> given by:
|
||||||
|
</p>
|
||||||
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
||||||
|
<span class="inlinemediaobject"><img src="../../../../equations/mapairy_ref1.svg"></span>
|
||||||
|
|
||||||
|
</p></blockquote></div>
|
||||||
|
<p>
|
||||||
|
The location parameter μ is the location of the distribution, while the scale
|
||||||
|
parameter [c] determines the width of the distribution. If the location
|
||||||
|
is zero, and the scale 1, then the result is a standard map-airy distribution.
|
||||||
|
</p>
|
||||||
|
<p>
|
||||||
|
The distribution describes the probability distribution of the area under
|
||||||
|
a Brownian excursion over a unit interval.
|
||||||
|
</p>
|
||||||
|
<p>
|
||||||
|
The following graph shows how the distributions moves as the location parameter
|
||||||
|
changes:
|
||||||
|
</p>
|
||||||
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
||||||
|
<span class="inlinemediaobject"><img src="../../../../graphs/mapairy_pdf1.svg" align="middle"></span>
|
||||||
|
|
||||||
|
</p></blockquote></div>
|
||||||
|
<p>
|
||||||
|
While the following graph shows how the shape (scale) parameter alters
|
||||||
|
the distribution:
|
||||||
|
</p>
|
||||||
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
||||||
|
<span class="inlinemediaobject"><img src="../../../../graphs/mapairy_pdf2.svg" align="middle"></span>
|
||||||
|
|
||||||
|
</p></blockquote></div>
|
||||||
|
<h5>
|
||||||
|
<a name="math_toolkit.dist_ref.dists.mapairy_dist.h0"></a>
|
||||||
|
<span class="phrase"><a name="math_toolkit.dist_ref.dists.mapairy_dist.member_functions"></a></span><a class="link" href="mapairy_dist.html#math_toolkit.dist_ref.dists.mapairy_dist.member_functions">Member
|
||||||
|
Functions</a>
|
||||||
|
</h5>
|
||||||
|
<pre class="programlisting"><span class="identifier">BOOST_MATH_GPU_ENABLED</span> <span class="identifier">mapairy_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">location</span> <span class="special">=</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">scale</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span>
|
||||||
|
</pre>
|
||||||
|
<p>
|
||||||
|
Constructs a mapairy distribution, with location parameter <span class="emphasis"><em>location</em></span>
|
||||||
|
and scale parameter <span class="emphasis"><em>scale</em></span>. When these parameters take
|
||||||
|
their default values (location = 0, scale = 1) then the result is a Standard
|
||||||
|
map-airy Distribution.
|
||||||
|
</p>
|
||||||
|
<p>
|
||||||
|
Requires scale > 0, otherwise calls <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
|
||||||
|
</p>
|
||||||
|
<pre class="programlisting"><span class="identifier">BOOST_MATH_GPU_ENABLED</span> <span class="identifier">RealType</span> <span class="identifier">location</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
|
||||||
|
</pre>
|
||||||
|
<p>
|
||||||
|
Returns the location parameter of the distribution.
|
||||||
|
</p>
|
||||||
|
<pre class="programlisting"><span class="identifier">BOOST_MATH_GPU_ENABLED</span> <span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
|
||||||
|
</pre>
|
||||||
|
<p>
|
||||||
|
Returns the scale parameter of the distribution.
|
||||||
|
</p>
|
||||||
|
<h5>
|
||||||
|
<a name="math_toolkit.dist_ref.dists.mapairy_dist.h1"></a>
|
||||||
|
<span class="phrase"><a name="math_toolkit.dist_ref.dists.mapairy_dist.non_member_accessors"></a></span><a class="link" href="mapairy_dist.html#math_toolkit.dist_ref.dists.mapairy_dist.non_member_accessors">Non-member
|
||||||
|
Accessors</a>
|
||||||
|
</h5>
|
||||||
|
<p>
|
||||||
|
All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
|
||||||
|
functions</a> that are generic to all distributions are supported:
|
||||||
|
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
|
||||||
|
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
|
||||||
|
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard Function</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard Function</a>,
|
||||||
|
__logcdf, __logpdf, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>,
|
||||||
|
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.median">median</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mode">mode</a>,
|
||||||
|
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.variance">variance</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a>,
|
||||||
|
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>,
|
||||||
|
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.range">range</a> and <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.support">support</a>. For this distribution
|
||||||
|
all non-member accessor functions are marked with <code class="computeroutput"><span class="identifier">BOOST_MATH_GPU_ENABLED</span></code>
|
||||||
|
and can be run on both host and device.
|
||||||
|
</p>
|
||||||
|
<p>
|
||||||
|
Note however that the map-airy distribution does not have a skewness, kurtosis,
|
||||||
|
etc. See <a class="link" href="../../pol_ref/assert_undefined.html" title="Mathematically Undefined Function Policies">mathematically
|
||||||
|
undefined function</a> to control whether these should fail to compile
|
||||||
|
with a BOOST_STATIC_ASSERTION_FAILURE, which is the default.
|
||||||
|
</p>
|
||||||
|
<p>
|
||||||
|
Alternately, the functions <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>,
|
||||||
|
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a> and
|
||||||
|
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>
|
||||||
|
will all return a <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
|
||||||
|
if called.
|
||||||
|
</p>
|
||||||
|
<p>
|
||||||
|
The domain of the random variable is [-[max_value], +[min_value]].
|
||||||
|
</p>
|
||||||
|
<h5>
|
||||||
|
<a name="math_toolkit.dist_ref.dists.mapairy_dist.h2"></a>
|
||||||
|
<span class="phrase"><a name="math_toolkit.dist_ref.dists.mapairy_dist.accuracy"></a></span><a class="link" href="mapairy_dist.html#math_toolkit.dist_ref.dists.mapairy_dist.accuracy">Accuracy</a>
|
||||||
|
</h5>
|
||||||
|
<p>
|
||||||
|
The error is within 4 epsilon except for the rapidly decaying left tail.
|
||||||
|
</p>
|
||||||
|
<p>
|
||||||
|
Errors in the PDF at 64-bit double precision:
|
||||||
|
</p>
|
||||||
|
<p>
|
||||||
|
<span class="inlinemediaobject"><img src="../../../../graphs/mapairy_pdf_accuracy_64.png"></span>
|
||||||
|
</p>
|
||||||
|
<p>
|
||||||
|
Errors in the CDF at 64-bit double precision:
|
||||||
|
</p>
|
||||||
|
<p>
|
||||||
|
<span class="inlinemediaobject"><img src="../../../../graphs/mapairy_cdf_accuracy_64.png"></span>
|
||||||
|
</p>
|
||||||
|
<p>
|
||||||
|
Errors in the CDF-complement at 64-bit double precision:
|
||||||
|
</p>
|
||||||
|
<p>
|
||||||
|
<span class="inlinemediaobject"><img src="../../../../graphs/mapairy_ccdf_accuracy_64.png"></span>
|
||||||
|
</p>
|
||||||
|
<h5>
|
||||||
|
<a name="math_toolkit.dist_ref.dists.mapairy_dist.h3"></a>
|
||||||
|
<span class="phrase"><a name="math_toolkit.dist_ref.dists.mapairy_dist.implementation"></a></span><a class="link" href="mapairy_dist.html#math_toolkit.dist_ref.dists.mapairy_dist.implementation">Implementation</a>
|
||||||
|
</h5>
|
||||||
|
<p>
|
||||||
|
See references.
|
||||||
|
</p>
|
||||||
|
<h5>
|
||||||
|
<a name="math_toolkit.dist_ref.dists.mapairy_dist.h4"></a>
|
||||||
|
<span class="phrase"><a name="math_toolkit.dist_ref.dists.mapairy_dist.references"></a></span><a class="link" href="mapairy_dist.html#math_toolkit.dist_ref.dists.mapairy_dist.references">References</a>
|
||||||
|
</h5>
|
||||||
|
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
|
||||||
|
<li class="listitem">
|
||||||
|
<a href="https://mathworld.wolfram.com/Map-AiryDistribution.html" target="_top">Wolfram
|
||||||
|
MathWorld: Map-Airy Distribution</a>
|
||||||
|
</li>
|
||||||
|
<li class="listitem">
|
||||||
|
T. Yoshimura, Numerical Evaluation and High Precision Approximation
|
||||||
|
Formula for Map-Airy Distribution, DOI: 10.36227/techrxiv.172053942.27675733/v1,
|
||||||
|
2024.
|
||||||
|
</li>
|
||||||
|
</ul></div>
|
||||||
|
</div>
|
||||||
|
<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
|
||||||
|
Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
|
||||||
|
Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
|
||||||
|
Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
|
||||||
|
Walker and Xiaogang Zhang<p>
|
||||||
|
Distributed under the Boost Software License, Version 1.0. (See accompanying
|
||||||
|
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
|
||||||
|
</p>
|
||||||
|
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|
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|
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|
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|
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|
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199
doc/html/math_toolkit/dist_ref/dists/saspoint5_dist.html
Normal file
199
doc/html/math_toolkit/dist_ref/dists/saspoint5_dist.html
Normal file
@@ -0,0 +1,199 @@
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|
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||||||
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<link rel="prev" href="rayleigh.html" title="Rayleigh Distribution">
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||||||
|
<a name="math_toolkit.dist_ref.dists.saspoint5_dist"></a><a class="link" href="saspoint5_dist.html" title="SαS Point5 Distribution">SαS Point5
|
||||||
|
Distribution</a>
|
||||||
|
</h4></div></div></div>
|
||||||
|
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">saspoint5</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span></pre>
|
||||||
|
<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span>
|
||||||
|
<span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a> <span class="special">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy<></a> <span class="special">></span>
|
||||||
|
<span class="keyword">class</span> <span class="identifier">saspoint5_distribution</span><span class="special">;</span>
|
||||||
|
|
||||||
|
<span class="keyword">typedef</span> <span class="identifier">saspoint5_distribution</span><span class="special"><></span> <span class="identifier">saspoint5</span><span class="special">;</span>
|
||||||
|
|
||||||
|
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
|
||||||
|
<span class="keyword">class</span> <span class="identifier">saspoint5_distribution</span>
|
||||||
|
<span class="special">{</span>
|
||||||
|
<span class="keyword">public</span><span class="special">:</span>
|
||||||
|
<span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span>
|
||||||
|
<span class="keyword">typedef</span> <span class="identifier">Policy</span> <span class="identifier">policy_type</span><span class="special">;</span>
|
||||||
|
|
||||||
|
<span class="identifier">BOOST_MATH_GPU_ENABLED</span> <span class="identifier">saspoint5_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">location</span> <span class="special">=</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">scale</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span>
|
||||||
|
|
||||||
|
<span class="identifier">BOOST_MATH_GPU_ENABLED</span> <span class="identifier">RealType</span> <span class="identifier">location</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
|
||||||
|
<span class="identifier">BOOST_MATH_GPU_ENABLED</span> <span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
|
||||||
|
<span class="special">};</span>
|
||||||
|
</pre>
|
||||||
|
<p>
|
||||||
|
It is special case of a <a href="http://en.wikipedia.org/wiki/Stable_distribution" target="_top">stable
|
||||||
|
distribution</a> with shape parameter α=1/2, β=0.
|
||||||
|
</p>
|
||||||
|
<p>
|
||||||
|
<a href="http://en.wikipedia.org/wiki/Probability_distribution" target="_top">probability
|
||||||
|
distribution function PDF</a> given by:
|
||||||
|
</p>
|
||||||
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
||||||
|
<span class="inlinemediaobject"><img src="../../../../equations/saspoint5_ref1.svg"></span>
|
||||||
|
|
||||||
|
</p></blockquote></div>
|
||||||
|
<p>
|
||||||
|
The location parameter μ is the location of the distribution, while the scale
|
||||||
|
parameter [c] determines the width of the distribution. If the location
|
||||||
|
is zero, and the scale 1, then the result is a standard SαS Point5 distribution.
|
||||||
|
</p>
|
||||||
|
<p>
|
||||||
|
This distribution has heavier tails than the Cauchy distribution.
|
||||||
|
</p>
|
||||||
|
<p>
|
||||||
|
The following graph shows how the distributions moves as the location parameter
|
||||||
|
changes:
|
||||||
|
</p>
|
||||||
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
||||||
|
<span class="inlinemediaobject"><img src="../../../../graphs/saspoint5_pdf1.svg" align="middle"></span>
|
||||||
|
|
||||||
|
</p></blockquote></div>
|
||||||
|
<p>
|
||||||
|
While the following graph shows how the shape (scale) parameter alters
|
||||||
|
the distribution:
|
||||||
|
</p>
|
||||||
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
||||||
|
<span class="inlinemediaobject"><img src="../../../../graphs/saspoint5_pdf2.svg" align="middle"></span>
|
||||||
|
|
||||||
|
</p></blockquote></div>
|
||||||
|
<h5>
|
||||||
|
<a name="math_toolkit.dist_ref.dists.saspoint5_dist.h0"></a>
|
||||||
|
<span class="phrase"><a name="math_toolkit.dist_ref.dists.saspoint5_dist.member_functions"></a></span><a class="link" href="saspoint5_dist.html#math_toolkit.dist_ref.dists.saspoint5_dist.member_functions">Member
|
||||||
|
Functions</a>
|
||||||
|
</h5>
|
||||||
|
<pre class="programlisting"><span class="identifier">BOOST_MATH_GPU_ENABLED</span> <span class="identifier">saspoint5_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">location</span> <span class="special">=</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">scale</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span>
|
||||||
|
</pre>
|
||||||
|
<p>
|
||||||
|
Constructs a SαS Point5 distribution, with location parameter <span class="emphasis"><em>location</em></span>
|
||||||
|
and scale parameter <span class="emphasis"><em>scale</em></span>. When these parameters take
|
||||||
|
their default values (location = 0, scale = 1) then the result is a Standard
|
||||||
|
SαS Point5 Distribution.
|
||||||
|
</p>
|
||||||
|
<p>
|
||||||
|
Requires scale > 0, otherwise calls <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
|
||||||
|
</p>
|
||||||
|
<pre class="programlisting"><span class="identifier">BOOST_MATH_GPU_ENABLED</span> <span class="identifier">RealType</span> <span class="identifier">location</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
|
||||||
|
</pre>
|
||||||
|
<p>
|
||||||
|
Returns the location parameter of the distribution.
|
||||||
|
</p>
|
||||||
|
<pre class="programlisting"><span class="identifier">BOOST_MATH_GPU_ENABLED</span> <span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
|
||||||
|
</pre>
|
||||||
|
<p>
|
||||||
|
Returns the scale parameter of the distribution.
|
||||||
|
</p>
|
||||||
|
<h5>
|
||||||
|
<a name="math_toolkit.dist_ref.dists.saspoint5_dist.h1"></a>
|
||||||
|
<span class="phrase"><a name="math_toolkit.dist_ref.dists.saspoint5_dist.non_member_accessors"></a></span><a class="link" href="saspoint5_dist.html#math_toolkit.dist_ref.dists.saspoint5_dist.non_member_accessors">Non-member
|
||||||
|
Accessors</a>
|
||||||
|
</h5>
|
||||||
|
<p>
|
||||||
|
All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
|
||||||
|
functions</a> that are generic to all distributions are supported:
|
||||||
|
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
|
||||||
|
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
|
||||||
|
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard Function</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard Function</a>,
|
||||||
|
__logcdf, __logpdf, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>,
|
||||||
|
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.median">median</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mode">mode</a>,
|
||||||
|
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.variance">variance</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a>,
|
||||||
|
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>,
|
||||||
|
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.range">range</a> and <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.support">support</a>. For this distribution
|
||||||
|
all non-member accessor functions are marked with <code class="computeroutput"><span class="identifier">BOOST_MATH_GPU_ENABLED</span></code>
|
||||||
|
and can be run on both host and device.
|
||||||
|
</p>
|
||||||
|
<p>
|
||||||
|
Note however that the SαS Point5 distribution does not have a mean, standard
|
||||||
|
deviation, etc. See <a class="link" href="../../pol_ref/assert_undefined.html" title="Mathematically Undefined Function Policies">mathematically
|
||||||
|
undefined function</a> to control whether these should fail to compile
|
||||||
|
with a BOOST_STATIC_ASSERTION_FAILURE, which is the default.
|
||||||
|
</p>
|
||||||
|
<p>
|
||||||
|
Alternately, the functions <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>,
|
||||||
|
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
|
||||||
|
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.variance">variance</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a>
|
||||||
|
and <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>
|
||||||
|
will all return a <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
|
||||||
|
if called.
|
||||||
|
</p>
|
||||||
|
<p>
|
||||||
|
The domain of the random variable is [-[max_value], +[min_value]].
|
||||||
|
</p>
|
||||||
|
<h5>
|
||||||
|
<a name="math_toolkit.dist_ref.dists.saspoint5_dist.h2"></a>
|
||||||
|
<span class="phrase"><a name="math_toolkit.dist_ref.dists.saspoint5_dist.accuracy"></a></span><a class="link" href="saspoint5_dist.html#math_toolkit.dist_ref.dists.saspoint5_dist.accuracy">Accuracy</a>
|
||||||
|
</h5>
|
||||||
|
<p>
|
||||||
|
The error is within 4 epsilon.
|
||||||
|
</p>
|
||||||
|
<p>
|
||||||
|
Errors in the PDF at 64-bit double precision:
|
||||||
|
</p>
|
||||||
|
<p>
|
||||||
|
<span class="inlinemediaobject"><img src="../../../../graphs/saspoint5_pdf_accuracy_64.png"></span>
|
||||||
|
</p>
|
||||||
|
<p>
|
||||||
|
Errors in the CDF-complement at 64-bit double precision:
|
||||||
|
</p>
|
||||||
|
<p>
|
||||||
|
<span class="inlinemediaobject"><img src="../../../../graphs/saspoint5_ccdf_accuracy_64.png"></span>
|
||||||
|
</p>
|
||||||
|
<h5>
|
||||||
|
<a name="math_toolkit.dist_ref.dists.saspoint5_dist.h3"></a>
|
||||||
|
<span class="phrase"><a name="math_toolkit.dist_ref.dists.saspoint5_dist.implementation"></a></span><a class="link" href="saspoint5_dist.html#math_toolkit.dist_ref.dists.saspoint5_dist.implementation">Implementation</a>
|
||||||
|
</h5>
|
||||||
|
<p>
|
||||||
|
See references.
|
||||||
|
</p>
|
||||||
|
<h5>
|
||||||
|
<a name="math_toolkit.dist_ref.dists.saspoint5_dist.h4"></a>
|
||||||
|
<span class="phrase"><a name="math_toolkit.dist_ref.dists.saspoint5_dist.references"></a></span><a class="link" href="saspoint5_dist.html#math_toolkit.dist_ref.dists.saspoint5_dist.references">References</a>
|
||||||
|
</h5>
|
||||||
|
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem">
|
||||||
|
T. Yoshimura, Numerical Evaluation and High Precision Approximation
|
||||||
|
Formula for SαS Point5 Distribution, DOI: 10.36227/techrxiv.172055253.37208198/v1,
|
||||||
|
2024.
|
||||||
|
</li></ul></div>
|
||||||
|
</div>
|
||||||
|
<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
|
||||||
|
Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
|
||||||
|
Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
|
||||||
|
Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
|
||||||
|
Walker and Xiaogang Zhang<p>
|
||||||
|
Distributed under the Boost Software License, Version 1.0. (See accompanying
|
||||||
|
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
|
||||||
|
</p>
|
||||||
|
</div>
|
||||||
|
<hr>
|
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|
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|
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|
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|
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66
doc/html/math_toolkit/double_exponential/gpu_usage.html
Normal file
66
doc/html/math_toolkit/double_exponential/gpu_usage.html
Normal file
@@ -0,0 +1,66 @@
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|
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|
||||||
|
<head>
|
||||||
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<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
|
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<link rel="up" href="../double_exponential.html" title="Double-exponential quadrature">
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|
<link rel="prev" href="de_caveats.html" title="Caveats">
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<td align="center"><a href="../../../../../../index.html">Home</a></td>
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<td align="center"><a href="../../../../../../libs/libraries.htm">Libraries</a></td>
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||||||
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|
||||||
|
<div class="section">
|
||||||
|
<div class="titlepage"><div><div><h3 class="title">
|
||||||
|
<a name="math_toolkit.double_exponential.gpu_usage"></a><a class="link" href="gpu_usage.html" title="GPU Usage">GPU Usage</a>
|
||||||
|
</h3></div></div></div>
|
||||||
|
<pre class="programlisting"> <span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">quadrature</span><span class="special">/</span><span class="identifier">exp_sinh</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
|
||||||
|
|
||||||
|
<span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">quadrature</span> <span class="special">{</span>
|
||||||
|
|
||||||
|
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Real</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Policy</span> <span class="special">=</span> <span class="identifier">policies</span><span class="special">::</span><span class="identifier">policy</span><span class="special"><></span> <span class="special">></span>
|
||||||
|
<span class="identifier">__device__</span> <span class="keyword">auto</span> <span class="identifier">exp_sinh_integrate</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">F</span><span class="special">&</span> <span class="identifier">f</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">tolerance</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">*</span> <span class="identifier">error</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">*</span> <span class="identifier">L1</span><span class="special">,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">size_t</span><span class="special">*</span> <span class="identifier">levels</span><span class="special">)</span>
|
||||||
|
|
||||||
|
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Real</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Policy</span> <span class="special">=</span> <span class="identifier">policies</span><span class="special">::</span><span class="identifier">policy</span><span class="special"><></span> <span class="special">></span>
|
||||||
|
<span class="identifier">__device__</span> <span class="keyword">auto</span> <span class="identifier">exp_sinh_integrate</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">F</span><span class="special">&</span> <span class="identifier">f</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">tolerance</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">*</span> <span class="identifier">error</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">*</span> <span class="identifier">L1</span><span class="special">,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">size_t</span><span class="special">*</span> <span class="identifier">levels</span><span class="special">)</span>
|
||||||
|
|
||||||
|
<span class="special">}}}</span>
|
||||||
|
</pre>
|
||||||
|
<p>
|
||||||
|
Quadrature is additionally able to run on CUDA (NVCC and NVRTC) platforms.
|
||||||
|
The major difference is outlined in the above function signatures. When used
|
||||||
|
on device these are free standing functions instead of using OOP like on
|
||||||
|
the host. The tables of abscissas and weights are stored in shared read only
|
||||||
|
memory on the device instead of being initialized when the class is constructed.
|
||||||
|
An example use case would be in the finite elements method computing a stiffness
|
||||||
|
matrix since it would consist of many different functions.
|
||||||
|
</p>
|
||||||
|
</div>
|
||||||
|
<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
|
||||||
|
Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
|
||||||
|
Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
|
||||||
|
Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
|
||||||
|
Walker and Xiaogang Zhang<p>
|
||||||
|
Distributed under the Boost Software License, Version 1.0. (See accompanying
|
||||||
|
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
|
||||||
|
</p>
|
||||||
|
</div>
|
||||||
|
<hr>
|
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|
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|
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121
doc/html/math_toolkit/gpu.html
Normal file
121
doc/html/math_toolkit/gpu.html
Normal file
@@ -0,0 +1,121 @@
|
|||||||
|
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|
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|
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|
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|
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<link rel="up" href="../overview.html" title="Chapter 1. Overview">
|
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<link rel="prev" href="error_handling.html" title="Error Handling">
|
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|
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|
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<td align="center"><a href="../../../../../index.html">Home</a></td>
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<td align="center"><a href="../../../../../libs/libraries.htm">Libraries</a></td>
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|
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|
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|
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|
||||||
|
</div>
|
||||||
|
<div class="section">
|
||||||
|
<div class="titlepage"><div><div><h2 class="title" style="clear: both">
|
||||||
|
<a name="math_toolkit.gpu"></a><a class="link" href="gpu.html" title="Support for GPU programming in Boost.Math">Support for GPU programming in Boost.Math</a>
|
||||||
|
</h2></div></div></div>
|
||||||
|
<h5>
|
||||||
|
<a name="math_toolkit.gpu.h0"></a>
|
||||||
|
<span class="phrase"><a name="math_toolkit.gpu.gpu_support"></a></span><a class="link" href="gpu.html#math_toolkit.gpu.gpu_support">GPU
|
||||||
|
Support</a>
|
||||||
|
</h5>
|
||||||
|
<p>
|
||||||
|
Selected functions, distributions, tools, etc. support running on both host
|
||||||
|
and devices. These functions will have the annotation <code class="computeroutput"><span class="identifier">BOOST_MATH_GPU_ENABLED</span></code>
|
||||||
|
or <code class="computeroutput"><span class="identifier">BOOST_MATH_CUDA_ENABLED</span></code>
|
||||||
|
next to their individual documentation. Functions marked with <code class="computeroutput"><span class="identifier">BOOST_MATH_GPU_ENABLED</span></code> are tested using CUDA
|
||||||
|
(both NVCC and NVRTC) as well as SYCL to provide a wide range of support. Functions
|
||||||
|
marked with <code class="computeroutput"><span class="identifier">BOOST_MATH_CUDA_ENABLED</span></code>
|
||||||
|
are few, but due to its restrictions SYCL is unsupported.
|
||||||
|
</p>
|
||||||
|
<h5>
|
||||||
|
<a name="math_toolkit.gpu.h1"></a>
|
||||||
|
<span class="phrase"><a name="math_toolkit.gpu.policies"></a></span><a class="link" href="gpu.html#math_toolkit.gpu.policies">Policies</a>
|
||||||
|
</h5>
|
||||||
|
<p>
|
||||||
|
The default policy on all devices is ignore error due to the lack of throwing
|
||||||
|
ability. A user can specify their own policy like usual, but when the code
|
||||||
|
is run on device it will be ignored.
|
||||||
|
</p>
|
||||||
|
<h5>
|
||||||
|
<a name="math_toolkit.gpu.h2"></a>
|
||||||
|
<span class="phrase"><a name="math_toolkit.gpu.how_to_build_with_device_support"></a></span><a class="link" href="gpu.html#math_toolkit.gpu.how_to_build_with_device_support">How
|
||||||
|
to build with device support</a>
|
||||||
|
</h5>
|
||||||
|
<p>
|
||||||
|
When compiling with CUDA or SYCL you will have to ensure that your code is
|
||||||
|
being run inside of a kernel function. It is not enough to simply compile existing
|
||||||
|
code with the NVCC compiler to run the code on the device. A simple CUDA kernel
|
||||||
|
to run the Beta Distribution CDF on NVCC would be:
|
||||||
|
</p>
|
||||||
|
<pre class="programlisting"><span class="identifier">__global__</span> <span class="keyword">void</span> <span class="identifier">cuda_beta_dist</span><span class="special">(</span><span class="keyword">const</span> <span class="keyword">double</span><span class="special">*</span> <span class="identifier">in</span><span class="special">,</span> <span class="keyword">double</span><span class="special">*</span> <span class="identifier">out</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">num_elements</span><span class="special">)</span>
|
||||||
|
<span class="special">{</span>
|
||||||
|
<span class="keyword">const</span> <span class="keyword">int</span> <span class="identifier">i</span> <span class="special">=</span> <span class="identifier">blockDim</span><span class="special">.</span><span class="identifier">x</span> <span class="special">*</span> <span class="identifier">blockIdx</span><span class="special">.</span><span class="identifier">x</span> <span class="special">+</span> <span class="identifier">threadIdx</span><span class="special">.</span><span class="identifier">x</span><span class="special">;</span>
|
||||||
|
|
||||||
|
<span class="keyword">if</span> <span class="special">(</span><span class="identifier">i</span> <span class="special"><</span> <span class="identifier">num_elements</span><span class="special">)</span>
|
||||||
|
<span class="special">{</span>
|
||||||
|
<span class="identifier">out</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">beta_distribution</span><span class="special"><</span><span class="keyword">double</span><span class="special">>(),</span> <span class="identifier">in</span><span class="special">[</span><span class="identifier">i</span><span class="special">]);</span>
|
||||||
|
<span class="special">}</span>
|
||||||
|
<span class="special">}</span>
|
||||||
|
</pre>
|
||||||
|
<p>
|
||||||
|
And on CUDA on NVRTC:
|
||||||
|
</p>
|
||||||
|
<pre class="programlisting"><span class="keyword">const</span> <span class="keyword">char</span><span class="special">*</span> <span class="identifier">cuda_kernel</span> <span class="special">=</span> <span class="identifier">R</span><span class="string">"(
|
||||||
|
#include <boost/math/distributions/beta.hpp>
|
||||||
|
extern "</span><span class="identifier">C</span><span class="string">" __global__
|
||||||
|
void test_beta_dist_kernel(const double* in, double* out, int num_elements)
|
||||||
|
{
|
||||||
|
const int i = blockDim.x * blockIdx.x + threadIdx.x;
|
||||||
|
if (i < num_elements)
|
||||||
|
{
|
||||||
|
out[i] = boost::math::cdf(boost::math::beta_distribution<double>(), in[i]);
|
||||||
|
}
|
||||||
|
}
|
||||||
|
)"</span><span class="special">;</span>
|
||||||
|
</pre>
|
||||||
|
<p>
|
||||||
|
And lastly on SYCL:
|
||||||
|
</p>
|
||||||
|
<pre class="programlisting"><span class="keyword">void</span> <span class="identifier">sycl_beta_dist</span><span class="special">(</span><span class="keyword">const</span> <span class="keyword">double</span><span class="special">*</span> <span class="identifier">in</span><span class="special">,</span> <span class="keyword">double</span><span class="special">*</span> <span class="identifier">out</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">num_elements</span><span class="special">,</span> <span class="identifier">sycl</span><span class="special">::</span><span class="identifier">queue</span><span class="special">&</span> <span class="identifier">q</span><span class="special">)</span>
|
||||||
|
<span class="special">{</span>
|
||||||
|
<span class="identifier">q</span><span class="special">.</span><span class="identifier">submit</span><span class="special">([&](</span><span class="identifier">sycl</span><span class="special">::</span><span class="identifier">handler</span><span class="special">&</span> <span class="identifier">h</span><span class="special">)</span> <span class="special">{</span>
|
||||||
|
<span class="identifier">h</span><span class="special">.</span><span class="identifier">parallel_for</span><span class="special">(</span><span class="identifier">sycl</span><span class="special">::</span><span class="identifier">range</span><span class="special"><</span><span class="number">1</span><span class="special">>(</span><span class="identifier">num_elements</span><span class="special">),</span> <span class="special">[=](</span><span class="identifier">sycl</span><span class="special">::</span><span class="identifier">id</span><span class="special"><</span><span class="number">1</span><span class="special">></span> <span class="identifier">i</span><span class="special">)</span> <span class="special">{</span>
|
||||||
|
<span class="identifier">out</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">beta_distribution</span><span class="special"><</span><span class="keyword">double</span><span class="special">>(),</span> <span class="identifier">in</span><span class="special">[</span><span class="identifier">i</span><span class="special">]);</span>
|
||||||
|
<span class="special">});</span>
|
||||||
|
<span class="special">});</span>
|
||||||
|
<span class="special">}</span>
|
||||||
|
</pre>
|
||||||
|
<p>
|
||||||
|
Once your kernel function has been written then use the framework mechanism
|
||||||
|
for launching the kernel.
|
||||||
|
</p>
|
||||||
|
</div>
|
||||||
|
<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
|
||||||
|
Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
|
||||||
|
Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
|
||||||
|
Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
|
||||||
|
Walker and Xiaogang Zhang<p>
|
||||||
|
Distributed under the Boost Software License, Version 1.0. (See accompanying
|
||||||
|
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
|
||||||
|
</p>
|
||||||
|
</div>
|
||||||
|
<hr>
|
||||||
|
<div class="spirit-nav">
|
||||||
|
<a accesskey="p" href="error_handling.html"><img src="../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../overview.html"><img src="../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../index.html"><img src="../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="compilers_overview.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a>
|
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|
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|
||||||
|
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|
||||||
|
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|
||||||
49
doc/html/math_toolkit/logistic.html
Normal file
49
doc/html/math_toolkit/logistic.html
Normal file
@@ -0,0 +1,49 @@
|
|||||||
|
<html>
|
||||||
|
<head>
|
||||||
|
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
|
||||||
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|
||||||
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|
||||||
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|
||||||
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|
||||||
|
<link rel="up" href="../special.html" title="Chapter 8. Special Functions">
|
||||||
|
<link rel="prev" href="sf_poly/jacobi.html" title="Jacobi Polynomials">
|
||||||
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<link rel="next" href="logistic/logit.html" title="logit">
|
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||||||
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<table cellpadding="2" width="100%"><tr>
|
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|
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|
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<td align="center"><a href="../../../../../libs/libraries.htm">Libraries</a></td>
|
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|
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|
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|
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|
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|
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|
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|
||||||
|
<div class="section">
|
||||||
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|
||||||
|
<a name="math_toolkit.logistic"></a><a class="link" href="logistic.html" title="Logistic Functions">Logistic Functions</a>
|
||||||
|
</h2></div></div></div>
|
||||||
|
<div class="toc"><dl class="toc">
|
||||||
|
<dt><span class="section"><a href="logistic/logit.html">logit</a></span></dt>
|
||||||
|
<dt><span class="section"><a href="logistic/logistic_sigmoid.html">logistic_sigmoid</a></span></dt>
|
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|
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|
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|
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|
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<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
|
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|
Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
|
||||||
|
Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
|
||||||
|
Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
|
||||||
|
Walker and Xiaogang Zhang<p>
|
||||||
|
Distributed under the Boost Software License, Version 1.0. (See accompanying
|
||||||
|
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
|
||||||
|
</p>
|
||||||
|
</div>
|
||||||
|
<hr>
|
||||||
|
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|
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|
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|
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|
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71
doc/html/math_toolkit/logistic/logistic_sigmoid.html
Normal file
71
doc/html/math_toolkit/logistic/logistic_sigmoid.html
Normal file
@@ -0,0 +1,71 @@
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<div class="section">
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<div class="titlepage"><div><div><h3 class="title">
|
||||||
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<a name="math_toolkit.logistic.logistic_sigmoid"></a><a class="link" href="logistic_sigmoid.html" title="logistic_sigmoid">logistic_sigmoid</a>
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||||||
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</h3></div></div></div>
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||||||
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<h5>
|
||||||
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<a name="math_toolkit.logistic.logistic_sigmoid.h0"></a>
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||||||
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<span class="phrase"><a name="math_toolkit.logistic.logistic_sigmoid.synopsis"></a></span><a class="link" href="logistic_sigmoid.html#math_toolkit.logistic.logistic_sigmoid.synopsis">Synopsis</a>
|
||||||
|
</h5>
|
||||||
|
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">logistic_sigmoid</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
|
||||||
|
|
||||||
|
<span class="keyword">namespace</span> <span class="identifier">boost</span> <span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span> <span class="special">{</span>
|
||||||
|
|
||||||
|
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">typename</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">Policy</span><span class="special">></span>
|
||||||
|
<span class="identifier">RealType</span> <span class="identifier">logistic_sigmoid</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Policy</span><span class="special">&);</span>
|
||||||
|
|
||||||
|
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">typename</span> <span class="identifier">RealType</span><span class="special">></span>
|
||||||
|
<span class="identifier">RealType</span> <span class="identifier">logistic_sigmoid</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">x</span><span class="special">)</span>
|
||||||
|
|
||||||
|
<span class="special">}}</span> <span class="comment">// namespaces</span>
|
||||||
|
</pre>
|
||||||
|
<h5>
|
||||||
|
<a name="math_toolkit.logistic.logistic_sigmoid.h1"></a>
|
||||||
|
<span class="phrase"><a name="math_toolkit.logistic.logistic_sigmoid.description"></a></span><a class="link" href="logistic_sigmoid.html#math_toolkit.logistic.logistic_sigmoid.description">Description</a>
|
||||||
|
</h5>
|
||||||
|
<p>
|
||||||
|
Returns the <a href="https://en.wikipedia.org/wiki/Logistic_function" target="_top">logistic
|
||||||
|
sigmoid function</a> This function is broadly useful, and is used to
|
||||||
|
compute the CDF of the logistic distribution. It is also sometimes referred
|
||||||
|
to as expit as it is the inverse of the logit function.
|
||||||
|
</p>
|
||||||
|
</div>
|
||||||
|
<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
|
||||||
|
Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
|
||||||
|
Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
|
||||||
|
Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
|
||||||
|
Walker and Xiaogang Zhang<p>
|
||||||
|
Distributed under the Boost Software License, Version 1.0. (See accompanying
|
||||||
|
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
|
||||||
|
</p>
|
||||||
|
</div>
|
||||||
|
<hr>
|
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<div class="spirit-nav">
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<a accesskey="p" href="logit.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../logistic.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="../bessel.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
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</html>
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70
doc/html/math_toolkit/logistic/logit.html
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70
doc/html/math_toolkit/logistic/logit.html
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<meta name="generator" content="DocBook XSL Stylesheets Vsnapshot">
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<link rel="home" href="../../index.html" title="Math Toolkit 4.2.1">
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<link rel="up" href="../logistic.html" title="Logistic Functions">
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<link rel="prev" href="../logistic.html" title="Logistic Functions">
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<table cellpadding="2" width="100%"><tr>
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<div class="section">
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|
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<a name="math_toolkit.logistic.logit"></a><a class="link" href="logit.html" title="logit">logit</a>
|
||||||
|
</h3></div></div></div>
|
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|
<h5>
|
||||||
|
<a name="math_toolkit.logistic.logit.h0"></a>
|
||||||
|
<span class="phrase"><a name="math_toolkit.logistic.logit.synopsis"></a></span><a class="link" href="logit.html#math_toolkit.logistic.logit.synopsis">Synopsis</a>
|
||||||
|
</h5>
|
||||||
|
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">logit</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
|
||||||
|
|
||||||
|
<span class="keyword">namespace</span> <span class="identifier">boost</span> <span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span> <span class="special">{</span>
|
||||||
|
|
||||||
|
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">typename</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">Policy</span><span class="special">></span>
|
||||||
|
<span class="identifier">RealType</span> <span class="identifier">logit</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Policy</span><span class="special">&);</span>
|
||||||
|
|
||||||
|
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">typename</span> <span class="identifier">RealType</span><span class="special">></span>
|
||||||
|
<span class="identifier">RealType</span> <span class="identifier">logit</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">x</span><span class="special">)</span>
|
||||||
|
|
||||||
|
<span class="special">}}</span> <span class="comment">// namespaces</span>
|
||||||
|
</pre>
|
||||||
|
<h5>
|
||||||
|
<a name="math_toolkit.logistic.logit.h1"></a>
|
||||||
|
<span class="phrase"><a name="math_toolkit.logistic.logit.description"></a></span><a class="link" href="logit.html#math_toolkit.logistic.logit.description">Description</a>
|
||||||
|
</h5>
|
||||||
|
<p>
|
||||||
|
Returns the <a href="https://en.wikipedia.org/wiki/Logit" target="_top">logit function</a>
|
||||||
|
This function is broadly useful, and is used to compute the Quantile of the
|
||||||
|
logistic distribution. The inverse of this function is the logistic_sigmoid.
|
||||||
|
</p>
|
||||||
|
</div>
|
||||||
|
<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
|
||||||
|
Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
|
||||||
|
Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
|
||||||
|
Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
|
||||||
|
Walker and Xiaogang Zhang<p>
|
||||||
|
Distributed under the Boost Software License, Version 1.0. (See accompanying
|
||||||
|
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
|
||||||
|
</p>
|
||||||
|
</div>
|
||||||
|
<hr>
|
||||||
|
<div class="spirit-nav">
|
||||||
|
<a accesskey="p" href="../logistic.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../logistic.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="logistic_sigmoid.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
|
||||||
|
</div>
|
||||||
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</body>
|
||||||
|
</html>
|
||||||
@@ -774,6 +774,7 @@ and as a CD ISBN 0-9504833-2-X 978-0-9504833-2-0, Classification 519.2-dc22.
|
|||||||
[include quadrature/wavelet_transforms.qbk]
|
[include quadrature/wavelet_transforms.qbk]
|
||||||
[include differentiation/numerical_differentiation.qbk]
|
[include differentiation/numerical_differentiation.qbk]
|
||||||
[include differentiation/autodiff.qbk]
|
[include differentiation/autodiff.qbk]
|
||||||
|
[include differentiation/autodiff_reverse.qbk]
|
||||||
[include differentiation/lanczos_smoothing.qbk]
|
[include differentiation/lanczos_smoothing.qbk]
|
||||||
[endmathpart]
|
[endmathpart]
|
||||||
|
|
||||||
|
|||||||
Reference in New Issue
Block a user