E407 Summation of Chebyshev Series

Function subprograms CHSUM and DCHSUM compute, for real arguments x in the specified intervals, one of the following four sums:

$S(x)$	$\sum$n=0NcnTn(x)	$(-1\le x\le 1)$	$(1)$
$S(x)$	$\sum$n=0NcnT2n(x)	$(-1\le x\le 1)$	$(2)$
$S(x)$	$\sum$n=0NcnT2n+1(x)	$(-1\le x\le 1)$	$(3)$
$S(x)$	$\sum$n=0NcnTn*(x)	$(0\le x\le 1)$	$(4)$

where $T$_n(x) is the Chebyshev polynomial of degree n and $T$_n^*(x) = T_n(2x - 1) .

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

Structure:

FUNCTION subprograms
User Entry Names: CHSUM, DCHSUM

Usage:

In any arithmetic expression, CHSUM(MODE,C,N,X) or DCHSUM(MODE,C,N,X)

has the value of the sum selected by MODE. CHSUM is of type REAL, and DCHSUM is of type DOUBLE PRECISION. C and X have the same type as the function name. MODE and N are of type INTEGER.

MODE

Type of sum to be evaluated $(MODE\; =\; 1,2,3,4)$ .

C

One-dimensional array with dimension (0:d), $d\; \ge N$ , containing the coefficients
$c$₀,c₁,...,c_N .

N

Limit N of summation.

X

Argument x.

Notes:

Note that some authors use a different definition for the constant term in (1), (2) and (4), i.e. $c$₀/2 instead of $c$₀ . Here, the definition of Ref. 1 is used.

References:

1. Y.L. Luke, Mathematical functions and their approximations, (Academic Press, New York 1975)
2. C.W. Clenshaw, Chebyshev series for mathematical functions, Mathematical Tables, Vol.5 (National Physical Laboratory, London, 1962).

E408