Sun Jan 3 23:45:44 MET 1999
REDUCE 3.7, 15-Jan-99 ...
1: 1:
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3: 3: % polydiv.tst -*- REDUCE -*-
% Test and demonstration file for enhanced polynomial division
% file polydiv.red.
% F.J.Wright@Maths.QMW.ac.uk, 7 Nov 1995.
% The example from "Computer Algebra" by Davenport, Siret & Tournier,
% first edition, section 2.3.3.
% First check that remainder still works as before.
% Compute the gcd of the polynomials a and b by Euclid's algorithm:
a := aa := x^8 + x^6 - 3x^4 - 3x^3 + 8x^2 + 2x - 5;
8 6 4 3 2
a := aa := x + x - 3*x - 3*x + 8*x + 2*x - 5
b := bb := 3x^6 + 5x^4 - 4x^2 - 9x + 21;
6 4 2
b := bb := 3*x + 5*x - 4*x - 9*x + 21
on rational;
off allfac;
c := remainder(a, b);
5 4 1 2 1
c := - ---*x + ---*x - ---
9 9 3
a := b$
b := c$
c := remainder(a, b);
117 2 441
c := - -----*x - 9*x + -----
25 25
a := b$
b := c$
c := remainder(a, b);
233150 102500
c := --------*x - --------
19773 6591
a := b$
b := c$
c := remainder(a, b);
1288744821
c := - ------------
543589225
a := b$
b := c$
c := remainder(a, b);
c := 0
off rational;
% Repeat using pseudo-remainders, to avoid rational arithmetic:
a := aa;
8 6 4 3 2
a := x + x - 3*x - 3*x + 8*x + 2*x - 5
b := bb;
6 4 2
b := 3*x + 5*x - 4*x - 9*x + 21
c := pseudo_remainder(a, b);
4 2
c := - 15*x + 3*x - 9
a := b$
b := c$
c := pseudo_remainder(a, b);
2
c := 15795*x + 30375*x - 59535
a := b$
b := c$
c := pseudo_remainder(a, b);
c := 1254542875143750*x - 1654608338437500
a := b$
b := c$
c := pseudo_remainder(a, b);
c := 12593338795500743100931141992187500
a := b$
b := c$
c := pseudo_remainder(a, b);
c := 0
% Example from Chris Herssens <herc@sulu.luc.ac.be>
% involving algebraic numbers in the coefficient ring
% (for which naive pseudo-division fails in REDUCE):
factor x;
a:=8*(15*sqrt(2)*x**3 + 18*sqrt(2)*x**2 + 10*sqrt(2)*x + 12*sqrt(2) -
5*x**4 - 6*x**3 - 30*x**2 - 36*x);
4 3 2
a := - 40*x + x *(120*sqrt(2) - 48) + x *(144*sqrt(2) - 240)
+ x*(80*sqrt(2) - 288) + 96*sqrt(2)
b:= - 16320*sqrt(2)*x**3 - 45801*sqrt(2)*x**2 - 50670*sqrt(2)*x -
26534*sqrt(2) + 15892*x**3 + 70920*x**2 + 86352*x + 24780;
3 2
b := x *( - 16320*sqrt(2) + 15892) + x *( - 45801*sqrt(2) + 70920)
+ x*( - 50670*sqrt(2) + 86352) - 26534*sqrt(2) + 24780
pseudo_remainder(a, b, x);
2 3/2
x *( - 51343372800*2 + 72663731640*2 + 106394745600*sqrt(2) - 152808065280) +
3/2
x*( - 77924736000*2 + 111722451600*2 + 167518488000*sqrt(2) - 236076547200)
3/2
- 26395315200*2 + 21508247760*2 + 58006274400*sqrt(2) - 51393323520
% Note: We must specify the division variable even though the
% polynomials are apparently univariate:
pseudo_remainder(a, b);
*** Main division variable selected is 2**(1/2)
7 6 5 4 3 2
652800*x + 708360*x - 2656800*x - 2660160*x + 4017600*x + 3676320*x
- 2630400*x - 2378880
% Confirm that quotient * b + remainder = constant * a:
pseudo_divide(a, b, x);
{x*(652800*sqrt(2) - 635680) - 1958400*2 + 858360*sqrt(2) + 2073984,
2 3/2
x *( - 51343372800*2 + 72663731640*2 + 106394745600*sqrt(2) - 152808065280)
+ x
3/2
*( - 77924736000*2 + 111722451600*2 + 167518488000*sqrt(2) - 236076547200)
3/2
- 26395315200*2 + 21508247760*2 + 58006274400*sqrt(2) - 51393323520}
first ws * b + second ws;
4
x *(20748595200*sqrt(2) - 31409618560)
3
+ x *(119127169920*sqrt(2) - 162183113472)
2
+ x *(237566198016*sqrt(2) - 337847596800)
+ x*(212209122560*sqrt(2) - 309143634432) + 75383084544*sqrt(2) - 99593256960
ws / a;
4 3
(x *(2593574400*sqrt(2) - 3926202320) + x *(14890896240*sqrt(2) - 20272889184)
2
+ x *(29695774752*sqrt(2) - 42230949600)
+ x*(26526140320*sqrt(2) - 38642954304) + 9422885568*sqrt(2) - 12449157120)/(
4 3 2
- 5*x + x *(15*sqrt(2) - 6) + x *(18*sqrt(2) - 30) + x*(10*sqrt(2) - 36)
+ 12*sqrt(2))
% is this constant?
on rationalize;
ws;
- 518714880*sqrt(2) + 785240464
% yes, it is constant
off rationalize;
on allfac;
remfac x;
procedure test_pseudo_division(a, b, x);
begin scalar qr, L;
qr := pseudo_divide(a, b, x);
L := lcof(b,x);
%% For versions of REDUCE prior to 3.6 use:
%% L := if b freeof x then b else lcof(b,x);
if first qr * b + second qr =
L^(deg(a,x)-deg(b,x)+1) * a then
write "Pseudo-division OK"
else
write "Pseudo-division failed"
end;
test_pseudo_division
a := 5x^4 + 4x^3 + 3x^2 + 2x + 1;
4 3 2
a := 5*x + 4*x + 3*x + 2*x + 1
test_pseudo_division(a, x, x);
Pseudo-division OK
test_pseudo_division(a, x^3, x);
Pseudo-division OK
test_pseudo_division(a, x^5, x);
Pseudo-division OK
test_pseudo_division(a, x^3 + x, x);
Pseudo-division OK
test_pseudo_division(a, 0, x);
***** Zero divisor
% intentional error!
test_pseudo_division(a, 1, x);
Pseudo-division OK
test_pseudo_division(5x^3 + 7y^2, 2x - y, x);
Pseudo-division OK
test_pseudo_division(5x^3 + 7y^2, 2x - y, y);
Pseudo-division OK
end;
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Time for test: 80 ms
5: 5:
Quitting
Sun Jan 3 23:45:45 MET 1999