E409 Summation of Trigonometric Series

Function subprograms RTRGSM and DTRGSM compute the sum of the trigonometric series

$f(x)\; =\; a$₀+∑_k=1ⁿa_kcoskx+∑_k=1^mb_ksinkx

for a given argument x in the range $-\pi \le x\; \le \pi$ and given coefficients $a$_k,b_k .

On CDC and Cray computers, the double-precision version DTRGSM is not available.

Structure:

FUNCTION subprogram
User Entry Names: RTRGSM, DTRGSM

Usage:

In any arithmetic expression, for $t=R$ (type REAL), $t=D$ (type DOUBLE PRECISION),

    tTRGSM(X,A,N,B,M,IOP)

X

(Type according to t) Argument x.

A

(Type according to t) One-dimensional array of dimension (0:d) where $d\; \ge N$ , containing the constant coefficient $a$₀ in A(0) and the cosine coefficients $a$_k (k=1,...,n) in A(k).

N

( INTEGER) The number n of cosine coefficients.

B

(Type according to t) One-dimensional array of length $\ge M$ , containing the sine coefficients $b$_k (k=1,...,n) in B(k).

M

( INTEGER) The number m of sine coefficients.

IOP

( INTEGER) An option number:
$=\; 1:$ the general case,
$=\; 2:$ all $b$_k are zero, i.e. $f(x)=f(-x)$ ,
$=\; 3:$ all $a$_k are zero, i.e. $f(x)=-f(-x)$ .

Method:

Standard recurrence relations are used for calculating the sum (see Ref. 1).

Notes:

For a function $f(z)$ given in the range $a\; \le z\; \le b$ , use the transformation

$x$	$=$
$x$	$=$

References:

1. W. Clenshaw, A note on the summation of Chebyshev series, MTAC (later renamed Math. Comp.) 9 (1955) 118--120.

Matrices, Vectors and Linear Equations

F001