vpasum

Numerical summation using variable precision

Since R2020b

Syntax

s = vpasum(f,a,b)

s = vpasum(f,x,a,b)

Description

s = vpasum(f,a,b) numerically approximates the sum of a series defined by f from a to b. The default summation variable x in f is determined by symvar. The summation bounds a and b must be real.

vpasum(f,[a,b]) is equal to vpasum(f,a,b).

s = vpasum(f,x,a,b) performs numerical summation using the summation variable x.

Examples

Numerically Sum Symbolic Expression

Numerically sum the symbolic expression $$\frac{1}{{1.2}^x}$$ from 1 to 1,000.

syms x;
s = vpasum(1/1.2^x,1,1000)

s = $$5.0$$

Summation of Symbolic Function

Find the summation of the symbolic function $$y(x)=x^2$$ from $$a$$ to $$b$$. Since the limits of summation must be real values, assume that the bounds $$a$$ and $$b$$ are real.

syms y(x)
syms a b real
y(x) = x^2;
s = vpasum(y,a,b)

s = 
$${\sum }_{x=a}^b{x}^2$$

Increase Performance of Numerical Series Summation

Compare the computation time to evaluate symbolic and numerical summations.

Find the symbolic summation of the series $\sum _{\mathit{k}=1}^{\infty }{\left(-1\right)}^{\mathit{k}}\frac{\log \left(\mathit{k}\right)}{{\mathit{k}}^3}$ using symsum. Use vpa to numerically evaluate the symbolic summation using 32 significant digits. Measure the time required to declare the symbolic summation and evaluate its numerical value.

syms k
tic
y = symsum((-1)^k*log(k)/k^3,k,1,Inf)

y = 
$${\sum }_{k=1}^{\infty }\frac{{\left(-1\right)}^k\hspace{0.17em}\log \left(k\right)}{k^3}$$

yVpa = vpa(y)

yVpa = $$0.059705906160195358363429266287926$$

toc

Elapsed time is 2.473831 seconds.

To increase computation performance (shorten computation time), use vpasum to evaluate the same numerical summation without evaluating the symbolic summation.

tic
y = vpasum((-1)^k*log(k)/k^3,k,1,Inf)

y = $$0.059705906160195358363429266287926$$

toc

Elapsed time is 0.094551 seconds.

Input Arguments

f — Expression or function to sum

symbolic number | symbolic variable | symbolic function | symbolic expression | symbolic vector | symbolic matrix | symbolic multidimensional array

Expression or function to sum, specified as a symbolic number, variable, function, expression, vector, matrix, or multidimensional array.

a,b — Limits of summation

two comma-separated numbers | symbolic numbers | symbolic variables | symbolic functions | symbolic expressions

Limits of summation, specified as two comma-separated numbers, symbolic numbers, symbolic variables, symbolic functions, or symbolic expressions. Specifying the summation range from a to b can also be done using a vector with two elements. The limits of summation must be real.

x — Summation variable

symbolic variable

Summation variable, specified as a symbolic variable. If x is not specified, the integration variable is determined by symvar(f).

Algorithms

Depending on whether the series is alternating or monotone, vpasum tries a number of strategies to calculate its limit: Levin's u-transformation, the Euler–Maclaurin formula, or van Wijngaarden's trick.

For example, the Euler–Maclaurin formula is

$${\displaystyle \sum _{i=a}^{b}f(i)=\frac{f(a)+f(b)}{2}}+{\displaystyle \underset{a}{\overset{b}{\int }}f(x)}dx+\left({\displaystyle \sum _{m=1}^{\lfloor p/2\rfloor }\frac{B_{2m}}{(2m)!}\left(f{(b)}^{2m-1}-f{(a)}^{2m-1}\right)}\right)+R_p,$$

where B_2m represents the 2mth Bernoulli number and R_p is an error term which depends on a, b, p, and f.

References

[1] Olver, F. W. J., A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds., Chapter 2.10 Sums and Sequences, NIST Digital Library of Mathematical Functions, Release 1.0.26 of 2020-03-15.

Version History

Introduced in R2020b