APPENDIX
BASIC PROGRAMS FOR THE MAIN SERIES
The following BASIC programs have been included as they give a good illustration of
how Chebyshev polynomials are used to produce the approximations to the functions
SIN, EXP, LN and ATN.
The series generator:
This subroutine is called by all the 'function' programs.
500 REM SERIES GENERATOR, ENTER
510 REM USING THE COUNTER BREG
520 REM AND ARRAY-A HOLDING THE
530 REM CONSTANTS.
540 REM FIRST VALUE IN Z.
550 LET M0=2*Z
560 LET M2=0
570 LET T=0
580 FOR I=BREG TO 1 STEP -1
590 LET M1=M2
600 LET U=T*M0-M2+A(BREG+1-I)
610 LET M2=T
620 LET T=U
630 NEXT I
640 LET T=T-M1
650 RETURN
660 REM LAST VALUE IN T.
In the above subroutine the variable are:
Z - the entry value.
T - the exit value.
M0 - mem-0
M1 - mem-1
M2 - mem-2
I - the counter for BREG.
U - a temporary variable for T.
A(1) to
A(BREG) - the constants.
BREG - the number of constants to be used.
To see how the Chebyshev polynomials are generated, record on paper the values of U,
M1, M2 and T through the lines 550 to 630, passing, say, 6 times through the loop, and
keeping the algebraic expressions for A(1) to A(6) without substituting numerical values.
Then record T-M1. The multipliers of the constants A(1) to A(6) will then be the re-
quired Chebyshev polynomials. More precisely, the multiplier of A(1) will be 2*T5(Z),
for A(2) it will be 2*T4(Z) and so on to 2*T1(Z) for A(5) and finally T0(Z) for A(6).
Note that T0(Z)=1, T1(Z)=Z and, for n>=2, Tn(Z)=2*Z*Tn-1(Z)-Tn-2(Z).