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* Jacobi Theta functions Implementations, tests, and ULP plotting programs are provided for the four Jacobi Theta functions per #373. Twenty-four public C++ functions are provided in all, covering various precision-preserving scenarios. Documentation for collaborators is provided in the code comments. Proper documentation for end users will be provided when the implementation and APIs are finalized. Some tests are failing; this implementation is meant to start a conversation. The core dilemma faced by the author was that large values of |q| resulted in slow convergence, and sometimes wildly inaccurate results. Following the implementation note in DLMF 20.14, I added code to switch over to the imaginary versions of the theta functions when |q| > 0.85. This restored accuracy such that all of the identity tests passed for a loose-enough epsilon, but then lost precision to the point that the Wolfram Alpha spot checks failed. It is the author's hope that someone with floating-point experience can tame the exponential dragons and squeeze the ULPs back down to a reasonable range when |q| is large. When #392 is merged I will add more thorough value tests, although I fully expect them to fail until the underlying precision issues are resolved. As a final note, the precision issues do not affect the z=0 case - the ULP plots indicate these return values within 2 ULP across all valid |q|. So that's a start. * [CI SKIP] Jacobi theta: Add special-value tests and more * Add tests covering z=0 special values from MathWorld * Add missing real_concept header * Replace M_PI and friends with constants::pi etc * Use BOOST_MATH_STD_USING in more places * Jacobi theta: Test two more of Watson's identities [CI SKIP] See https://mathworld.wolfram.com/JacobiThetaFunctions.html (Equations 48 and 49) * Improve precision of Jacobi theta functions [CI SKIP] Rewrite the private imaginary versions to use double-sided summations following DLMF 20.13.4 and 20.13.5. This cuts down the worst of the precision issues by a factor of 10, and gets more of the tests to pass. I am confident enough in the code path to eliminate the compile-time __JACOBI_THETA_USE_IMAGINARY flag. In fact the imaginary-z code paths are now enabled for all |q| > 0.04, i.e. most legal values of q. More extensive tests will be needed to illuminate any remaining precision issues. * Jacobi theta: Make changes suggested in #394 [CI SKIP] * Add LICENSE notice to main file * Document convergence criteria * Eliminate eps*eps = 0 logic. This causes some disagreement with the zero returned by Wolfram Alpha for z=0, q > 0.99 in the fourth function. Mathematically, the fourth function is never exactly zero, so I don't trust Wolfram here. * Per code-review comments, remove multiplications by floating-point 2. * Tweak the plotting programs to display their titles, and to uniformly use `float` as their CoarseType and `long double` as their `PreciseType`. * Add quadrature tests to Jacobi theta functions [CI SKIP] The quadrature tests revealed a problem in the m1 functions: they too should switch to the _IMAGINARY logic for q > exp(-pi), or will suffer from slow convergence. Fix them. Also tighten tolerances for many tests from sqrt(eps) to 100 * eps. * Test Jacobi thetas against elliptic functions and elliptic integrals [CI SKIP] See: * https://dlmf.nist.gov/22.2 * https://dlmf.nist.gov/20.9#i * Test Jacobi Thetas against their Laplace transforms [CI SKIP] See: * https://dlmf.nist.gov/20.10#ii I did find some disagreement, and dropped the negative sign from the theta1 equation. DLMF's theta2 and theta3 Laplace transform equations do not agree at all with the computed values - will need to investigate. In the meantime, the two implemented equations agree to 4 EPS so I am keeping them. * Add a note on using log1p with Jacobi theta functions [CI SKIP] See discussion: * https://github.com/boostorg/math/pull/394#issuecomment-655871762 * Add random data tests to Jacobi Theta functions [CI SKIP] Add a test data generator program for the Jacobi theta functions. This program will produce data for the tau parameterization, so that precision isn't lost during the log-transformation. This distinguishes it from the Wolfram Alpha data, which is parameterized by q. A few of these new random-data tests are failing, but not by obscene margins (< 100 EPS). These failures will be addressed when the test tolerances are finalized. * Add small-tau tests and simplify Jacobi Theta code [CI SKIP] Add tests for small tau (i.e. large q). The tests are failing with mean ~ 200 EPS and max ~ 800 EPS. These look like worst-case input, and should be the focus of future accuracy improvements. This commit also simplifies the _IMAGINARY code by abstracting all of the loops into a single svelte function. * Add user documentation for Jacobi Theta functions [CI SKIP] * Add function graphs to Jacobi Theta docs [CI SKIP] * Define Jacobi Theta test tolerances [CI SKIP] * Add implementation note on Jacobi theta functions [CI SKIP] * Consolidate Jacobi Theta ULPs plotting programs [CI SKIP] * Fix q domain checking of jacobi_theta4 [CI SKIP] * Add ULPs plots to Jacobi Theta docs [CI SKIP] Also add the built HTML files for easy evaluation. A full rebuild is needed for the new docs to appear in the indexes. * Add missing Jacobi Theta ULPs plots [CI SKIP] * Add LaTeX source for Jacobi Theta equations [CI SKIP] * Remove unused Jacobi Theta PNG equations [CI SKIP] * Add Jacobi Theta performance script [CI SKIP] Provided by @NAThompson. * Remove vestigial eps*eps check from jacobi_theta3 [CI SKIP] * Update Jacobi Theta docs per code review comments [CI SKIP] * Enable arg promotion for Jacobi Theta functions [CI SKIP] Add Jacobi theta functions to the instantiation tests and fix up everything needed to make them pass. This changes the function signatures to use promote_args. * Fix Jacobi Theta plotting script [CI SKIP] This script broke when the promote_args API was added. * Change Jacobi Theta convergence criterion [CI SKIP] Compare the non-oscillating part of the delta to the previous one. This avoids some headaches comparing the delta to the partial sum, because the partial sum can be a small number due to the oscillating component alternating signs. Because successive terms involve either q^n^2 or exp(-(pi*n)^2), convergence should still happen pretty quickly. Graphs have been updated and tests still passs with no noticeable difference.
372 lines
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372 lines
15 KiB
Plaintext
[/
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Copyright (c) 2020 Evan Miller
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Use, modification and distribution are subject to the
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Boost Software License, Version 1.0. (See accompanying file
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LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
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]
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[section:jacobi_theta Jacobi Theta Functions]
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[section:jacobi_theta_overview Overview of the Jacobi Theta Functions]
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The Jacobi Theta functions are a set of four inter-related periodic functions of /x/ which are expressed in terms of a parameter /q/ (also called the nome), or a closely related value, [tau]
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[footnote [@https://en.wikipedia.org/wiki/Theta_function Wikipedia: Theta function]]
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[footnote [@https://mathworld.wolfram.com/JacobiThetaFunctions.html Weisstein, Eric W. "Jacobi Theta Functions." From MathWorld - A Wolfram Web Resource.]]
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[footnote [@https://dlmf.nist.gov/20 Digital Library of Mathematical Functions: Theta Functions, Reinhardt, W. P., Walker, P. L.]].
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They are
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[equation jacobi_theta1] [/ \theta_1(x, q) := 2 \sum_{n=0}^\infty (-1)^n q^{(n+\frac{1}{2})^2} \sin((2n+1)x) ]
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[equation jacobi_theta2] [/ \theta_2(x, q) := 2 \sum_{n=0}^\infty q^{(n+\frac{1}{2})^2} \cos((2n+1)x) ]
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[equation jacobi_theta3] [/ \theta_3(x, q) := 1 + 2 \sum_{n=1}^\infty q^{n^2} \cos(2nx) ]
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[equation jacobi_theta4] [/ \theta_4(x, q) := 1 + 2 \sum_{n=1}^\infty (-1)^n q^{n^2} \cos(2nx) ]
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[graph jacobi_theta]
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Plots of the four theta functions for /q/=0.15.
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Appropriately multiplied and divided, these four theta functions can be used
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to implement the [link math_toolkit.jacobi.jac_over Jacobi elliptic functions]; but this is not really
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recommended, as the existing Boost implementations are likely faster and
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more accurate.
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Most applications will want to use the /q/ parameterization of the functions: `__jacobi_theta1`, `__jacobi_theta2`, `__jacobi_theta3`, and `__jacobi_theta4`, where /q/ is restricted to the domain (0, 1).
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These four functions are equivalent to Mathematica's [@https://reference.wolfram.com/language/ref/EllipticTheta.html EllipticTheta] function (whose first argument is the function number).
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A second [tau] parameterization is also provided for all four functions, where
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[equation jacobi_theta_nome] [/ q = \exp(i\pi\tau) ]
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Note that there is a slight difference between [tau] in the equation above and the `tau` in the Boost function signatures.
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The mathematical [tau] is assumed to be a purely imaginary number, but the Boost argument is real-valued.
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Boost treats its real-valued argument as an imaginary number; that is, it implicitly multiplies the argument by /i/.
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This assumption of [tau]'s imaginarity is not required by the mathematics, but it does cover the most common application domains.
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[heading Accuracy considerations]
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The purpose of the [tau] parameterization is to provide increased accuracy either when /q/ is expressible as an exponential or is very close to unity.
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For example, instead of:
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jacobi_theta1(x, exp(-a));
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A more accurate computation will take advantage of [tau]:
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jacobi_theta1tau(x, a / boost::math::constants::pi<T>());
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Internally, Boost implements the /q/ parameterization by taking the logarithm of /q/ and passing it to the [tau] parameterization; as such, using the [tau] parameterization directly will generally yield greater precision.
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As another example, if the complement of /q/ is known with great accuracy, then instead of:
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jacobi_theta1(x, 1-q_complement);
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It is more accurate to use `__log1p` and pass in the result to the [tau] version:
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jacobi_theta1tau(x, -boost::math::log1p(-q_complement) / boost::math::constants::pi<T>());
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Additional "minus 1" versions of the third and fourth theta functions are provided. Similar in spirit to `__expm1`, these functions return one less than the evaluated function, and yield increased accuracy when /q/ is small.
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[heading Testing]
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Results of the theta functions are tested against Wolfram Alpha data, as well as random values computed at high precision.
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In addition, the tests verify the majority of the identities described in [@https://dlmf.nist.gov/20.7 DLMF Chapter 20.7].
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[endsect] [/section:jacobi_theta_overview Overview of the Jacobi Theta Functions]
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[section:jacobi_theta1 Jacobi Theta Function [theta][sub 1]]
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[heading Synopsis]
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``
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#include <boost/math/special_functions/jacobi_theta.hpp>
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``
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namespace boost { namespace math {
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template <class T, class U>
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``__sf_result`` jacobi_theta1(T x, U q);
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template <class T, class U, class Policy>
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``__sf_result`` jacobi_theta1(T x, U q, const Policy&);
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template <class T, class U>
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``__sf_result`` jacobi_theta1tau(T x, U tau);
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template <class T, class U, class Policy>
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``__sf_result`` jacobi_theta1tau(T x, U tau, const Policy&);
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}} // namespaces
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[heading Description]
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The functions calculate the value of first [link math_toolkit.jacobi_theta.jacobi_theta_overview Jacobi Theta function], parameterized either in terms of the nome /q/:
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[equation jacobi_theta1] [/ \theta_1(x, q) := 2 \sum_{n=0}^\infty (-1)^n q^{(n+\frac{1}{2})^2} \sin((2n+1)x) ]
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Or in terms of an imaginary [tau]:
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[equation jacobi_theta1tau] [/ \theta_1(x|\tau) := 2 \sum_{n=0}^\infty (-1)^n \exp(i\pi\tau{(n+0.5)^2}) \sin((2n+1)x) ]
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The nome /q/ is restricted to the domain (0, 1), returning the result of __domain_error otherwise. The following graph shows the theta function at various values of /q/:
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[graph jacobi_theta1]
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[optional_policy]
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[heading Accuracy]
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The following [link math_toolkit.ulps_plots ULPs plot] is representative, fixing /q/=0.5 and varying /x/ from 0 to 2[pi]:
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[graph jacobi_theta1_float]
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The envelope represents the function's [@https://en.wikipedia.org/wiki/Condition_number#One_variable condition number].
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Note that relative accuracy degenerates periodically near [theta][sub 1]=0.
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Fixing /x/=5 and varying /q/, the ULPs plot looks like:
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[graph jacobi_theta1q_float]
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Accuracy tends to degenerate near /q/=1 (small [tau]).
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[heading Implementation]
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The /q/ parameterization is implemented using the [tau] parameterization, where [tau]=-log(/q/)/[pi].
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If [tau] is greater than or equal to 1, the summation above is used as-is.
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However if [tau] < 1, the following identity [@https://dlmf.nist.gov/20.7#viii DLMF 20.7.30] is used, defining [tau]'=-1/[tau]:
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[equation jacobi_theta1_imaginary] [/ (-i\tau)^{1/2}\theta_1(x|\tau)=-i\exp(i\tau'x^2/\pi)\theta_1(x\tau'|\tau') ]
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This transformation of variables ensures that the function will always converge in a small number of iterations.
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[endsect] [/section:jacobi_theta1 Jacobi Theta Function [theta][sub 1]]
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[section:jacobi_theta2 Jacobi Theta Function [theta][sub 2]]
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[heading Synopsis]
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``
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#include <boost/math/special_functions/jacobi_theta.hpp>
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``
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namespace boost { namespace math {
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template <class T, class U>
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``__sf_result`` jacobi_theta2(T x, U q);
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template <class T, class U, class Policy>
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``__sf_result`` jacobi_theta2(T x, U q, const Policy&);
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template <class T, class U>
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``__sf_result`` jacobi_theta2tau(T x, U tau);
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template <class T, class U, class Policy>
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``__sf_result`` jacobi_theta2tau(T x, U tau, const Policy&);
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}} // namespaces
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[heading Description]
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The functions calculate the value of second [link math_toolkit.jacobi_theta.jacobi_theta_overview Jacobi Theta function], parameterized either in terms of the nome /q/:
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[equation jacobi_theta2] [/ \theta_2(x, q) := 2 \sum_{n=0}^\infty q^{(n+\frac{1}{2})^2} \cos((2n+1)x) ]
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Or in terms of an imaginary [tau]:
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[equation jacobi_theta2tau] [/ \theta_2(x|\tau) := 2 \sum_{n=0}^\infty \exp(i\pi\tau{(n+0.5)^2}) \cos((2n+1)x) ]
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The nome /q/ is restricted to the domain (0, 1), returning the result of __domain_error otherwise.
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The following graph shows the theta function at various values of /q/:
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[graph jacobi_theta2]
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[optional_policy]
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[heading Accuracy]
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The following [link math_toolkit.ulps_plots ULPs plot] is representative, fixing /q/=0.5 and varying /x/ from 0 to 2[pi]:
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[graph jacobi_theta2_float]
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The envelope represents the function's [@https://en.wikipedia.org/wiki/Condition_number#One_variable condition number].
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Note that relative accuracy degenerates periodically near [theta][sub 2]=0.
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Fixing /x/=0.4 and varying /q/, the ULPs plot looks like:
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[graph jacobi_theta2q_float]
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Accuracy tends to degenerate near /q/=1 (small [tau]).
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[heading Implementation]
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The /q/ parameterization is implemented using the [tau] parameterization, where [tau]=-log(/q/)/[pi].
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If [tau] is greater than or equal to 1, the summation above is used as-is.
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However if [tau] < 1, the following identity [@https://dlmf.nist.gov/20.7#viii DLMF 20.7.31] is used, defining [tau]'=-1/[tau]:
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[equation jacobi_theta2_imaginary] [/ (-i\tau)^{1/2}\theta_2(x|\tau)=\exp(i\tau'x^2/\pi)\theta_4(x\tau'|\tau') ]
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This transformation of variables ensures that the function will always converge in a small number of iterations.
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[endsect] [/section:jacobi_theta2 Jacobi Theta Function [theta][sub 2]]
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[section:jacobi_theta3 Jacobi Theta Function [theta][sub 3]]
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[heading Synopsis]
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``
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#include <boost/math/special_functions/jacobi_theta.hpp>
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``
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namespace boost { namespace math {
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template <class T, class U>
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``__sf_result`` jacobi_theta3(T x, U q);
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template <class T, class U, class Policy>
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``__sf_result`` jacobi_theta3(T x, U q, const Policy&);
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template <class T, class U>
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``__sf_result`` jacobi_theta3tau(T x, U tau);
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template <class T, class U, class Policy>
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``__sf_result`` jacobi_theta3tau(T x, U tau, const Policy&);
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template <class T, class U>
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``__sf_result`` jacobi_theta3m1(T x, U q);
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template <class T, class U, class Policy>
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``__sf_result`` jacobi_theta3m1(T x, U q, const Policy&);
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template <class T, class U>
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``__sf_result`` jacobi_theta3m1tau(T x, U tau);
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template <class T, class U, class Policy>
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``__sf_result`` jacobi_theta3m1tau(T x, U tau, const Policy&);
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}} // namespaces
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[heading Description]
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The functions calculate the value of third [link math_toolkit.jacobi_theta.jacobi_theta_overview Jacobi Theta function], parameterized either in terms of the nome /q/:
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[equation jacobi_theta3] [/ \theta_3(x, q) := 1 + 2 \sum_{n=1}^\infty q^{n^2} \cos(2nx) ]
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Or in terms of an imaginary [tau]:
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[equation jacobi_theta3tau] [/ \theta_3(x|\tau) := 1 + 2 \sum_{n=1}^\infty \exp(i\pi\tau{n^2}) \cos(2nx) ]
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The nome /q/ is restricted to the domain (0, 1), returning the result of __domain_error otherwise.
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The following graph shows the theta function at various values of /q/:
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[graph jacobi_theta3]
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[optional_policy]
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A second quartet of functions (functions containing `m1`) compute one less than the value of the third theta function.
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These versions of the functions provide increased accuracy when the result is close to unity.
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[heading Accuracy]
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The following [link math_toolkit.ulps_plots ULPs plot] is representative, fixing /q/=0.5 and varying /x/ from 0 to 2[pi]:
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[graph jacobi_theta3_float]
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The envelope represents the function's [@https://en.wikipedia.org/wiki/Condition_number#One_variable condition number].
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Note that relative accuracy degenerates periodically near [theta][sub 3]=1.
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Fixing /x/=0.4 and varying /q/, the ULPs plot looks like:
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[graph jacobi_theta3q_float]
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Accuracy tends to degenerate near /q/=1 (small [tau]).
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[heading Implementation]
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The /q/ parameterization is implemented using the [tau] parameterization, where [tau]=-log(/q/)/[pi].
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If [tau] is greater than or equal to 1, the summation above is used as-is.
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However if [tau] < 1, the following identity [@https://dlmf.nist.gov/20.7#viii DLMF 20.7.32] is used, defining [tau]'=-1/[tau]:
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[equation jacobi_theta3_imaginary] [/ (-i\tau)^{1/2}\theta_3(x|\tau)=\exp(i\tau'x^2/\pi)\theta_3(x\tau'|\tau') ]
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This transformation of variables ensures that the function will always converge in a small number of iterations.
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[endsect] [/section:jacobi_theta3 Jacobi Theta Function [theta][sub 3]]
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[section:jacobi_theta4 Jacobi Theta Function [theta][sub 4]]
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[heading Synopsis]
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``
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#include <boost/math/special_functions/jacobi_theta.hpp>
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``
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namespace boost { namespace math {
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template <class T, class U>
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``__sf_result`` jacobi_theta4(T x, U q);
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template <class T, class U, class Policy>
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``__sf_result`` jacobi_theta4(T x, U q, const Policy&);
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template <class T, class U>
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``__sf_result`` jacobi_theta4tau(T x, U tau);
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template <class T, class U, class Policy>
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``__sf_result`` jacobi_theta4tau(T x, U tau, const Policy&);
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template <class T, class U>
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``__sf_result`` jacobi_theta4m1(T x, U q);
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template <class T, class U, class Policy>
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``__sf_result`` jacobi_theta4m1(T x, U q, const Policy&);
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template <class T, class U>
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``__sf_result`` jacobi_theta4m1tau(T x, U tau);
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template <class T, class U, class Policy>
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``__sf_result`` jacobi_theta4m1tau(T x, U tau, const Policy&);
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}} // namespaces
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[heading Description]
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The functions calculate the value of fourth [link math_toolkit.jacobi_theta.jacobi_theta_overview Jacobi Theta function], parameterized either in terms of the nome /q/:
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[equation jacobi_theta4] [/ \theta_4(x, q) := 1 + 2 \sum_{n=1}^\infty (-1)^n q^{n^2} \cos(2nx) ]
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Or in terms of an imaginary [tau]:
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[equation jacobi_theta4tau] [/ \theta_4(x|\tau) := 1 + 2 \sum_{n=1}^\infty (-1)^n \exp(i\pi\tau{n^2}) \cos(2nx) ]
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The nome /q/ is restricted to the domain (0, 1), returning the result of __domain_error otherwise.
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The following graph shows the theta function at various values of /q/:
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[graph jacobi_theta4]
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[optional_policy]
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A second quartet of functions (functions containing `m1`) compute one less than the value of the fourth theta function.
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These versions of the functions provide increased accuracy when the result is close to unity.
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[heading Accuracy]
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The following [link math_toolkit.ulps_plots ULPs plot] is representative, fixing /q/=0.5 and varying /x/ from 0 to 2[pi]:
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[graph jacobi_theta4_float]
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The envelope represents the function's [@https://en.wikipedia.org/wiki/Condition_number#One_variable condition number].
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Note that relative accuracy degenerates periodically near [theta][sub 4]=1.
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Fixing /x/=5 and varying /q/, the ULPs plot looks like:
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[graph jacobi_theta4q_float]
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Accuracy tends to degenerate near /q/=1 (small [tau]).
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[heading Implementation]
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The /q/ parameterization is implemented using the [tau] parameterization, where [tau]=-log(/q/)/[pi].
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If [tau] is greater than or equal to 1, the summation above is used as-is.
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However if [tau] < 1, the following identity [@https://dlmf.nist.gov/20.7#viii DLMF 20.7.33] is used, defining [tau]'=-1/[tau]:
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[equation jacobi_theta4_imaginary] [/ (-i\tau)^{1/2}\theta_4(x|\tau)=\exp(i\tau'x^2/\pi)\theta_2(x\tau'|\tau') ]
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This transformation of variables ensures that the function will always converge in a small number of iterations.
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[endsect] [/section:jacobi_theta4 Jacobi Theta Function [theta][sub 4]]
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[endsect] [/section:jacobi_theta Jacobi Theta Functions]
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