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Last revised: August 09, 2019 at 15:22:32 GMT
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A B C D E F G H I L M N O P Q R S T U V W
A B C D E F G H I L N O P R S T U V W
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This documentation aims to use of the following naming and formatting conventions. diff --git a/doc/html/math_toolkit/cubic_b.html b/doc/html/math_toolkit/cubic_b.html index 71c76ce82..f61d62a36 100644 --- a/doc/html/math_toolkit/cubic_b.html +++ b/doc/html/math_toolkit/cubic_b.html @@ -50,6 +50,8 @@ Real operator()(Real x) const; Real prime(Real x) const; + + Real double_prime(Real x) const; }; }} // namespaces @@ -60,7 +62,7 @@ B-Spline Interpolation
- The cubic B-spline class provided by boost allows fast and accurate interpolation + The cubic B-spline class provided by Boost allows fast and accurate interpolation of a function which is known at equally spaced points. The cubic B-spline interpolation is numerically stable as it uses compactly supported basis functions constructed via iterative convolution. This is to be contrasted to traditional cubic spline @@ -131,6 +133,20 @@ order lower than the guarantees on the original function, see Numerical Analysis, Graduate Texts in Mathematics, 181, Rainer Kress for details.
++ The last interesting member is the second derivative, evaluated via +
+double ypp = spline.double_prime(x); ++
+ The basis functions of the spline are cubic polynomials, so the second derivative + is simply linear interpolation. But the interpolation is not constrained by + data (you don't pass in the second derivatives), and hence is less accurate + than would be expected from assuming it is a linear interpolation. The problem + is especially pronounced at the boundaries, where the second derivative is + totally unconstrained. Use the second derivative of the cubic B-spline interpolator + only in desperation. +
Finally, note that this is an interpolator, not an extrapolator. Therefore, you should strenuously avoid evaluating the spline outside the endpoints. However, diff --git a/doc/html/math_toolkit/navigation.html b/doc/html/math_toolkit/navigation.html index 88799d994..ff37d1ef2 100644 --- a/doc/html/math_toolkit/navigation.html +++ b/doc/html/math_toolkit/navigation.html @@ -27,7 +27,7 @@ Navigation
Boost.Math documentation is provided in both HTML and PDF formats. diff --git a/doc/sf/hypergeometric.qbk b/doc/sf/hypergeometric.qbk index 115d8706a..4605486a1 100644 --- a/doc/sf/hypergeometric.qbk +++ b/doc/sf/hypergeometric.qbk @@ -1,4 +1,3 @@ - [section:hypergeometric Hypergeometric Functions] [section:hyper_geometric_1f0 Hypergeometric [sub 1]/F/[sub 0] ] @@ -513,11 +512,11 @@ error inherent in calculating the N'th term via logarithms. # Beals, Richard, and Roderick Wong. ['Special functions: a graduate text.] Vol. 126. Cambridge University Press, 2010. # Pearson, John W., Sheehan Olver, and Mason A. Porter. ['Numerical methods for the computation of the confluent and Gauss hypergeometric functions.] Numerical Algorithms 74.3 (2017): 821-866. # Luke, Yudell L. ['Algorithms for Rational Approximations for a Confluent Hypergeometric Function II.] MISSOURI UNIV KANSAS CITY DEPT OF MATHEMATICS, 1976. -# Derezinski, Jan. ['Hypergeometric type functions and their symmetries.] Annales Henri Poincaré. Vol. 15. No. 8. Springer Basel, 2014. +# Derezinski, Jan. ['Hypergeometric type functions and their symmetries.] Annales Henri Poincar[eacute]. Vol. 15. No. 8. Springer Basel, 2014. # Keith E. Muller ['Computing the confluent hypergeometric function, M(a, b, x)]. Numer. Math. 90: 179-196 (2001). # Carlo Morosi, Livio Pizzocchero. ['On the expansion of the Kummer function in terms of incomplete Gamma functions.] Arch. Inequal. Appl. 2 (2004), 49-72. # Jose Luis Lopez, Nico M. Temme. ['Asymptotics and numerics of polynomials used in Tricomi and Buchholz expansions of Kummer functions]. Numerische Mathematik, August 2010. -# Javier Sesma. ['The Temme’s sum rule for confluent hypergeometric functions revisited]. Journal of Computational and Applied Mathematics 163 (2004) 429–431. +# Javier Sesma. ['The Temme's sum rule for confluent hypergeometric functions revisited]. Journal of Computational and Applied Mathematics 163 (2004) 429-431. # Javier Segura, Nico M. Temme. ['Numerically satisfactory solutions of Kummer recurrence relations]. Numer. Math. (2008) 111:109-119. # Alfredo Deano, Javier Segura. ['Transitory Minimal Solutions Of Hypergeometric Recursions And Pseudoconvergence of Associated Continued Fractions]. Mathematics of Computation, Volume 76, Number 258, April 2007. # W. Gautschi. ['Computational aspects of three-term recurrence relations]. SIAM Review 9, no.1 (1967) 24-82.