bson@fruit-and-fibre.ai.mit.edu (Jan Brittenson) (10/21/90)
This is a general Taylor program. It returns the Taylor series
coinciding with a function at any given X. Not just at X=0 (i.e.
Maclaurin series), like the built-in function TAYLR. It uses the
SYSEVAL #546Dh to create the algebraic 'n!' where n is an arbitrary
real. The SYSEVAL #54AFh is used to obtain the function ! and the
short <2h>.
The usage is identical to TAYLR (is TAYLR differentiable?) with an
additional fourth argument, the X value.
Two examples - the former a Maclaurin series, the latter a Taylor
series at X=1:
Example 1.
'EXP(X)' 'X' 3 0 TAYLRX
--> '1+X/1!+X^2/2!+X^3/3!'
'EXP(X)' 'X' 3 TAYLR
--> '1+X+0.5*X^2+1/3!*X^3'
Example 2.
'LN(X)' 'X' 5 1 TAYLRX
--> '(X-1)/1!-(X-1)/2!+2*(X-1)^3/3!-6*(X-1)^4/4!+24*(X-1)^5/5!'
'LNT(X)' SWAP = DEFINE
1.25 LN 1.25 LNT -
--> -.000033532019
O /
\/
/\~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
O \
@ TAYLRX #8CD8h
@ General Taylor series
@ f: function of dv
@ dv: variable of differentiation
@ N: polynomial degree
@ pt: X value
@ FC: !
@ S3: <2h>
@ c: degree counter
%%HP: T(3)A(R)F(,);
\<< \-> f dv N pt
\<< '1!' # 54AFh SYSEVAL
3 ROLL DROP
dv pt 2 \->LIST
\-> FC S3 dv0
\<< f dv0 | @ start with f(pt)
1 N FOR c
f dv \.d 'f' STO @ f'(dv) -> f
f dv0 | dv pt - c ^ * @ 'f(pt)*(dv-pt)^c'
c FC S3 # 546Dh SYSEVAL / + @ 'c!' / ; sum up
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