Gaussian elimination: Difference between revisions
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{{wont work with|ELLA ALGOL 68|Any (with appropriate job cards) - tested with release [http://sourceforge.net/projects/algol68/files/algol68toc/algol68toc-1.8.8d/algol68toc-1.8-8d.fc9.i386.rpm/download 1.8-8d] - due to extensive use of '''format'''[ted]}} |
{{wont work with|ELLA ALGOL 68|Any (with appropriate job cards) - tested with release [http://sourceforge.net/projects/algol68/files/algol68toc/algol68toc-1.8.8d/algol68toc-1.8-8d.fc9.i386.rpm/download 1.8-8d] - due to extensive use of '''format'''[ted]}} |
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'''File: prelude_exception.a68'''<lang algol68># -*- coding: utf-8 -*- # |
'''File: prelude_exception.a68'''<lang algol68># -*- coding: utf-8 -*- # |
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COMMENT PROVIDES |
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⚫ | |||
MODE FIXED; INT fixed exception, unfixed exception; |
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PROC (STRING message) FIXED raise, raise value error |
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END COMMENT |
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# Note: ℵ indicates attribute is "private", and |
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⚫ | |||
should not be used outside of this prelude # |
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⚫ | |||
⚫ | |||
⚫ | |||
⚫ | |||
⚫ | |||
FIXED fixed exception = TRUE, unfixed exception = FALSE; |
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⚫ | |||
⚫ | |||
⚫ | |||
⚫ | |||
⚫ | |||
putf(stand error, ($"Exception:"$, $xg$, message, $l$)); |
putf(stand error, ($"Exception:"$, $xg$, message, $l$)); |
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stop |
stop |
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); |
); |
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PROC raise value error = (SIMPLEOUT message) |
PROC raise value error = (#ℵ#SIMPLEOUT message)FIXED: |
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IF raise(message) NE fixed exception THEN exception value error; FALSE FI; |
IF raise(message) NE fixed exception THEN exception value error; FALSE FI; |
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SKIP</lang>'''File: prelude_mat_lib.a68'''<lang algol68># -*- coding: utf-8 -*- # |
SKIP</lang>'''File: prelude_mat_lib.a68'''<lang algol68># -*- coding: utf-8 -*- # |
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COMMENT REQUIRES |
COMMENT PRELUDE REQUIRES |
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MODE SCAL |
MODE SCAL = REAL; |
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FORMAT scal repr |
FORMAT scal repr = real repr |
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# and various SCAL OPerators # |
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END COMMENT |
END COMMENT |
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COMMENT PRELUDE PROIVIDES |
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⚫ | |||
MODE VEC, MAT; |
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OP :=:, -:=, +:=, *:=, /:=; |
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FORMAT sub, sep, bus; |
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FORMAT vec repr, mat repr |
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END COMMENT |
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# Note: ℵ indicates attribute is "private", and |
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should not be used outside of this prelude # |
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⚫ | |||
INT #ℵ#lwb mat := 1, #ℵ#upb mat := 0; |
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MODE VEC = [lwb vec:upb vec]SCAL, |
MODE VEC = [lwb vec:upb vec]SCAL, |
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MAT = [lwb |
MAT = [lwb mat:upb mat,lwb vec:upb vec]SCAL; |
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FORMAT sub = $"("$, sep=$", "$, bus = $")"$; |
FORMAT sub := $"( "$, sep := $", "$, bus := $")"$, nl:=$lxx$; |
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FORMAT vec repr = $f(sub)n(upb vec - lwb vec)(f(scal repr)f(sep))f(scal repr)f(bus)$; |
FORMAT vec repr := $f(sub)n(upb vec - lwb vec)(f(scal repr)f(sep))f(scal repr)f(bus)$; |
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FORMAT mat repr := $f(sub)n(upb mat - lwb mat)(f( vec repr)f(nl))f( vec repr)f(bus)$; |
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# OPerators to swap the contents of two VECtors # |
# OPerators to swap the contents of two VECtors # |
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SKIP</lang>'''File: prelude_gaussian_elimination.a68'''<lang algol68># -*- coding: utf-8 -*- # |
SKIP</lang>'''File: prelude_gaussian_elimination.a68'''<lang algol68># -*- coding: utf-8 -*- # |
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COMMENT PRELUDE REQUIRES |
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MODE SCAL = REAL, |
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REAL near min scal = min real ** 0.99, |
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MODE VEC = []REAL, |
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MODE MAT = [,]REAL, |
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FORMAT scal repr = real repr, |
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and various OPerators of MAT and VEC |
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END COMMENT |
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COMMENT PRELUDE PROVIDES |
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PROC(MAT a, b)MAT gaussian elimination; |
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PROC(REF MAT a, b)REF MAT in situ gaussian elimination |
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END COMMENT |
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#################################################### |
#################################################### |
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# using Gaussian elimination, find x where A*x = b # |
# using Gaussian elimination, find x where A*x = b # |
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#################################################### |
#################################################### |
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PROC in situ gaussian elimination = (REF MAT a, |
PROC in situ gaussian elimination = (REF MAT a, b)REF MAT: ( |
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# Note: a and b are modified "in situ", and b is returned as x # |
# Note: a and b are modified "in situ", and b is returned as x # |
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); |
); |
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PROC gaussian elimination = (MAT in a, |
PROC gaussian elimination = (MAT in a, in b)MAT: ( |
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# Note: a and b are cloned and not modified "in situ" # |
# Note: a and b are cloned and not modified "in situ" # |
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[UPB in a, 2 UPB in a]SCAL a := in a; |
[UPB in a, 2 UPB in a]SCAL a := in a; |
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SKIP</lang>'''File: postlude_exception.a68'''<lang algol68># -*- coding: utf-8 -*- # |
SKIP</lang>'''File: postlude_exception.a68'''<lang algol68># -*- coding: utf-8 -*- # |
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COMMENT PRELUDE PROIVIDES |
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PROC VOID exception too many iterations, exception value error; |
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END COMMENT |
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SKIP EXIT |
SKIP EXIT |
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exception too many iterations: |
exception too many iterations: |
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PR READ "postlude_exception.a68" PR</lang>'''Output:''' |
PR READ "postlude_exception.a68" PR</lang>'''Output:''' |
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<pre> |
<pre> |
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(-.010000000000002, 1.602790394502130, -1.613203059905640, 1.245494121371510, -.490989719584686, .065760696175236) |
( -.010000000000002, 1.602790394502130, -1.613203059905640, 1.245494121371510, -.490989719584686, .065760696175236) |
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</pre> |
</pre> |
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Revision as of 03:19, 1 October 2012
Problem: Solve Ax=b using Gaussian elimination then backwards substitution. A being an n by n matrix. Also, x and b are n by 1 vectors. To improve accuracy, please use partial pivoting and scaling.
ALGOL 68
File: prelude_exception.a68<lang algol68># -*- coding: utf-8 -*- # COMMENT PROVIDES
MODE FIXED; INT fixed exception, unfixed exception; PROC (STRING message) FIXED raise, raise value error
END COMMENT
- Note: ℵ indicates attribute is "private", and
should not be used outside of this prelude #
MODE FIXED = BOOL; # if an exception is detected, can it be fixed "on-site"? # FIXED fixed exception = TRUE, unfixed exception = FALSE;
MODE #ℵ#SIMPLEOUTV = [0]UNION(CHAR, STRING, INT, REAL, BOOL, BITS); MODE #ℵ#SIMPLEOUTM = [0]#ℵ#SIMPLEOUTV; MODE #ℵ#SIMPLEOUTT = [0]#ℵ#SIMPLEOUTM; MODE SIMPLEOUT = [0]#ℵ#SIMPLEOUTT;
PROC raise = (#ℵ#SIMPLEOUT message)FIXED: (
putf(stand error, ($"Exception:"$, $xg$, message, $l$)); stop
);
PROC raise value error = (#ℵ#SIMPLEOUT message)FIXED:
IF raise(message) NE fixed exception THEN exception value error; FALSE FI;
SKIP</lang>File: prelude_mat_lib.a68<lang algol68># -*- coding: utf-8 -*- # COMMENT PRELUDE REQUIRES
MODE SCAL = REAL; FORMAT scal repr = real repr # and various SCAL OPerators #
END COMMENT
COMMENT PRELUDE PROIVIDES
MODE VEC, MAT; OP :=:, -:=, +:=, *:=, /:=; FORMAT sub, sep, bus; FORMAT vec repr, mat repr
END COMMENT
- Note: ℵ indicates attribute is "private", and
should not be used outside of this prelude #
INT #ℵ#lwb vec := 1, #ℵ#upb vec := 0; INT #ℵ#lwb mat := 1, #ℵ#upb mat := 0; MODE VEC = [lwb vec:upb vec]SCAL,
MAT = [lwb mat:upb mat,lwb vec:upb vec]SCAL;
FORMAT sub := $"( "$, sep := $", "$, bus := $")"$, nl:=$lxx$; FORMAT vec repr := $f(sub)n(upb vec - lwb vec)(f(scal repr)f(sep))f(scal repr)f(bus)$; FORMAT mat repr := $f(sub)n(upb mat - lwb mat)(f( vec repr)f(nl))f( vec repr)f(bus)$;
- OPerators to swap the contents of two VECtors #
PRIO =:= = 1; OP =:= = (REF VEC u, v)VOID:
FOR i TO UPB u DO SCAL scal=u[i]; u[i]:=v[i]; v[i]:=scal OD;
OP +:= = (REF VEC lhs, VEC rhs)REF VEC: (
FOR i TO UPB lhs DO lhs[i] +:= rhs[i] OD; lhs
);
OP -:= = (REF VEC lhs, VEC rhs)REF VEC: (
FOR i TO UPB lhs DO lhs[i] -:= rhs[i] OD; lhs
);
OP *:= = (REF VEC lhs, SCAL rhs)REF VEC: (
FOR i TO UPB lhs DO lhs[i] *:= rhs OD; lhs
);
OP /:= = (REF VEC lhs, SCAL rhs)REF VEC: (
SCAL inv = 1 / rhs; # multiplication is faster # FOR i TO UPB lhs DO lhs[i] *:= inv OD; lhs
);
SKIP</lang>File: prelude_gaussian_elimination.a68<lang algol68># -*- coding: utf-8 -*- # COMMENT PRELUDE REQUIRES
MODE SCAL = REAL, REAL near min scal = min real ** 0.99, MODE VEC = []REAL, MODE MAT = [,]REAL, FORMAT scal repr = real repr, and various OPerators of MAT and VEC
END COMMENT
COMMENT PRELUDE PROVIDES
PROC(MAT a, b)MAT gaussian elimination; PROC(REF MAT a, b)REF MAT in situ gaussian elimination
END COMMENT
- using Gaussian elimination, find x where A*x = b #
PROC in situ gaussian elimination = (REF MAT a, b)REF MAT: (
- Note: a and b are modified "in situ", and b is returned as x #
FOR diag TO UPB a-1 DO INT pivot row := diag; SCAL pivot factor := ABS a[diag,diag]; FOR row FROM diag + 1 TO UPB a DO # Full pivoting # SCAL abs a diag = ABS a[row,diag]; IF abs a diag>=pivot factor THEN pivot row := row; pivot factor := abs a diag FI OD; # now we have the "best" diag to full pivot, do the actual pivot # IF diag NE pivot row THEN # a[pivot row,] =:= a[diag,]; XXX: unoptimised # #DB# a[pivot row,diag:] =:= a[diag,diag:]; # XXX: optimised # b[pivot row,] =:= b[diag,] # swap/pivot the diags of a & b # FI;
IF ABS a[diag,diag] <= near min scal THEN raise value error("singular matrix") FI; SCAL a diag reciprocal := 1 / a[diag, diag];
FOR row FROM diag+1 TO UPB a DO SCAL factor = a[row,diag] * a diag reciprocal; # a[row,] -:= factor * a[diag,] XXX: "unoptimised" # #DB# a[row,diag+1:] -:= factor * a[diag,diag+1:];# XXX: "optimised" # b[row,] -:= factor * b[diag,] OD OD;
- We have a triangular matrix, at this point we can traverse backwards
up the diagonal calculating b\A Converting it initial to a diagonal matrix, then to the identity. #
FOR diag FROM UPB a BY -1 TO 1+LWB a DO
IF ABS a[diag,diag] <= near min scal THEN raise value error("Zero pivot encountered?") FI; SCAL a diag reciprocal = 1 / a[diag,diag];
FOR row TO diag-1 DO SCAL factor = a[row,diag] * a diag reciprocal; # a[row,diag] -:= factor * a[diag,diag]; XXX: "unoptimised" so remove # #DB# b[row,] -:= factor * b[diag,] OD;
- Now we have only diagonal elements we can simply divide b
by the values along the diagonal of A. # b[diag,] *:= a diag reciprocal OD;
b # EXIT #
);
PROC gaussian elimination = (MAT in a, in b)MAT: (
- Note: a and b are cloned and not modified "in situ" #
[UPB in a, 2 UPB in a]SCAL a := in a; [UPB in b, 2 UPB in b]SCAL b := in b; in situ gaussian elimination(a,b)
);
SKIP</lang>File: postlude_exception.a68<lang algol68># -*- coding: utf-8 -*- # COMMENT PRELUDE PROIVIDES
PROC VOID exception too many iterations, exception value error;
END COMMENT
SKIP EXIT exception too many iterations: exception value error:
stop</lang>File: test_Gaussian_elimination.a68<lang algol68>#!/usr/bin/algol68g-full --script #
- -*- coding: utf-8 -*- #
PR READ "prelude_exception.a68" PR;
- define the attributes of the scalar field being used #
MODE SCAL = REAL; FORMAT scal repr = $g(-0,real width)$;
- create "near min scal" as is scales better then small real #
SCAL near min scal = min real ** 0.99;
PR READ "prelude_mat_lib.a68" PR; PR READ "prelude_gaussian_elimination.a68" PR;
MAT a =(( 1.00, 0.00, 0.00, 0.00, 0.00, 0.00),
( 1.00, 0.63, 0.39, 0.25, 0.16, 0.10), ( 1.00, 1.26, 1.58, 1.98, 2.49, 3.13), ( 1.00, 1.88, 3.55, 6.70, 12.62, 23.80), ( 1.00, 2.51, 6.32, 15.88, 39.90, 100.28), ( 1.00, 3.14, 9.87, 31.01, 97.41, 306.02));
VEC b = (-0.01, 0.61, 0.91, 0.99, 0.60, 0.02);
[UPB b,1]SCAL col b; col b[,1]:= b;
upb vec := 2 UPB a;
printf((vec repr, gaussian elimination(a,col b)));
PR READ "postlude_exception.a68" PR</lang>Output:
( -.010000000000002, 1.602790394502130, -1.613203059905640, 1.245494121371510, -.490989719584686, .065760696175236)
C
This modifies A and b in place, which might not be quite desirable. <lang c>#include <stdio.h>
- include <stdlib.h>
- include <math.h>
- define mat_elem(a, y, x, n) (a + ((y) * (n) + (x)))
void swap_row(double *a, double *b, int r1, int r2, int n) { double tmp, *p1, *p2; int i;
if (r1 == r2) return; for (i = 0; i < n; i++) { p1 = mat_elem(a, r1, i, n); p2 = mat_elem(a, r2, i, n); tmp = *p1, *p1 = *p2, *p2 = tmp; } tmp = b[r1], b[r1] = b[r2], b[r2] = tmp; }
void gauss_eliminate(double *a, double *b, double *x, int n) {
- define A(y, x) (*mat_elem(a, y, x, n))
int i, j, col; double max, tmp;
for (col = 0; col < n; col++) { j = col, max = A(j, j);
for (i = col + 1; i < n; i++) if ((tmp = fabs(A(i, col))) > max) j = i, max = tmp;
swap_row(a, b, col, j, n);
for (i = col + 1; i < n; i++) { tmp = A(i, col) / A(col, col); for (j = col + 1; j < n; j++) A(i, j) -= tmp * A(col, j); A(i, col) = 0; b[i] -= tmp * b[col]; } } for (col = n - 1; col >= 0; col--) { tmp = b[col]; for (j = n - 1; j > col; j--) tmp -= x[j] * A(col, j); x[col] = tmp / A(col, col); }
- undef A
}
int main(void) { double a[] = { 1.00, 0.00, 0.00, 0.00, 0.00, 0.00, 1.00, 0.63, 0.39, 0.25, 0.16, 0.10, 1.00, 1.26, 1.58, 1.98, 2.49, 3.13, 1.00, 1.88, 3.55, 6.70, 12.62, 23.80, 1.00, 2.51, 6.32, 15.88, 39.90, 100.28, 1.00, 3.14, 9.87, 31.01, 97.41, 306.02 }; double b[] = { -0.01, 0.61, 0.91, 0.99, 0.60, 0.02 }; double x[6]; int i;
gauss_eliminate(a, b, x, 6);
for (i = 0; i < 6; i++) printf("%g\n", x[i]);
return 0;
}</lang>
- Output:
-0.01 1.60279 -1.6132 1.24549 -0.49099 0.0657607
J
%. , J's matrix divide verb, directly solves systems of determined and of over-determined linear equations directly. This example J session builds a noisy sine curve on the half circle, fits quintic and quadratic equations, and displays the results of evaluating these polynomials.
<lang J>
f=: 6j2&": NB. formatting verb
sin=: 1&o. NB. verb to evaluate circle function 1, the sine
add_noise=: ] + (* (_0.5 + 0 ?@:#~ #)) NB. AMPLITUDE add_noise SIGNAL
f RADIANS=: o.@:(%~ i.@:>:)5 NB. monadic circle function is pi times 0.00 0.63 1.26 1.88 2.51 3.14
f SINES=: sin RADIANS 0.00 0.59 0.95 0.95 0.59 0.00
f NOISY_SINES=: 0.1 add_noise SINES _0.01 0.61 0.91 0.99 0.60 0.02
A=: (^/ i.@:#) RADIANS NB. A is the quintic coefficient matrix
NB. display the equation to solve (f A) ; 'x' ; '=' ; f@:,. NOISY_SINES
┌────────────────────────────────────┬─┬─┬──────┐ │ 1.00 0.00 0.00 0.00 0.00 0.00│x│=│ _0.01│ │ 1.00 0.63 0.39 0.25 0.16 0.10│ │ │ 0.61│ │ 1.00 1.26 1.58 1.98 2.49 3.13│ │ │ 0.91│ │ 1.00 1.88 3.55 6.70 12.62 23.80│ │ │ 0.99│ │ 1.00 2.51 6.32 15.88 39.90100.28│ │ │ 0.60│ │ 1.00 3.14 9.87 31.01 97.41306.02│ │ │ 0.02│ └────────────────────────────────────┴─┴─┴──────┘
f QUINTIC_COEFFICIENTS=: NOISY_SINES %. A NB. %. solves the linear system _0.01 1.71 _1.88 1.48 _0.58 0.08
quintic=: QUINTIC_COEFFICIENTS&p. NB. verb to evaluate the polynomial
NB. %. also solves the least squares fit for overdetermined system quadratic=: (NOISY_SINES %. (^/ i.@:3:) RADIANS)&p. NB. verb to evaluate quadratic. quadratic
_0.0200630695393961729 1.26066877804926536 _0.398275112136019516&p.
NB. The quintic is agrees with the noisy data, as it should f@:(NOISY_SINES ,. sin ,. quadratic ,. quintic) RADIANS _0.01 0.00 _0.02 _0.01 0.61 0.59 0.61 0.61 0.91 0.95 0.94 0.91 0.99 0.95 0.94 0.99 0.60 0.59 0.63 0.60 0.02 0.00 0.01 0.02
f MID_POINTS=: (+ -:@:(-/@:(2&{.)))RADIANS _0.31 0.31 0.94 1.57 2.20 2.83
f@:(sin ,. quadratic ,. quintic) MID_POINTS _0.31 _0.46 _0.79 0.31 0.34 0.38 0.81 0.81 0.77 1.00 0.98 1.00 0.81 0.83 0.86 0.31 0.36 0.27
</lang>
Mathematica
<lang Mathematica>GaussianElimination[A_?MatrixQ, b_?VectorQ] := Last /@ RowReduce[Flatten /@ Transpose[{A, b}]]</lang>
MATLAB
<lang MATLAB> function [ x ] = GaussElim( A, b)
% Ensures A is n by n sz = size(A); if sz(1)~=sz(2)
fprintf('A is not n by n\n'); clear x; return;
end
n = sz(1);
% Ensures b is n x 1. if n~=sz(1)
fprintf('b is not 1 by n.\n'); return
end
x = zeros(n,1); aug = [A b]; tempmatrix = aug;
for i=2:sz(1)
% Find maximum of row and divide by the maximum tempmatrix(1,:) = tempmatrix(1,:)/max(tempmatrix(1,:)); % Finds the maximum in column temp = find(abs(tempmatrix) - max(abs(tempmatrix(:,1)))); if length(temp)>2 for j=1:length(temp)-1 if j~=temp(j) maxi = j; %maxi = column number of maximum break; end end else % length(temp)==2 maxi=1; end % Row swap if maxi is not 1 if maxi~=1 temp = tempmatrix(maxi,:); tempmatrix(maxi,:) = tempmatrix(1,:); tempmatrix(1,:) = temp; end % Row reducing for j=2:length(tempmatrix)-1 tempmatrix(j,:) = tempmatrix(j,:)-tempmatrix(j,1)/tempmatrix(1,1)*tempmatrix(1,:); if tempmatrix(j,j)==0 || isnan(tempmatrix(j,j)) || abs(tempmatrix(j,j))==Inf fprintf('Error: Matrix is singular.\n'); clear x; return end end aug(i-1:end,i-1:end) = tempmatrix; % Decrease matrix size tempmatrix = tempmatrix(2:end,2:end);
end
% Backwards Substitution x(end) = aug(end,end)/aug(end,end-1); for i=n-1:-1:1
x(i) = (aug(i,end)-dot(aug(i,1:end-1),x))/aug(i,i);
end
end </lang>
PARI/GP
If A and B have floating-point numbers (t_REAL
s) then the following uses Gaussian elimination:
<lang parigp>matsolve(A,B)</lang>
If the entries are integers, then p-adic lifting (Dixon 1982) is used instead.
Tcl
<lang tcl>package require math::linearalgebra
set A {
{1.00 0.00 0.00 0.00 0.00 0.00} {1.00 0.63 0.39 0.25 0.16 0.10} {1.00 1.26 1.58 1.98 2.49 3.13} {1.00 1.88 3.55 6.70 12.62 23.80} {1.00 2.51 6.32 15.88 39.90 100.28} {1.00 3.14 9.87 31.01 97.41 306.02}
} set b {-0.01 0.61 0.91 0.99 0.60 0.02} puts -nonewline [math::linearalgebra::show [math::linearalgebra::solveGauss $A $b] "%.2f"]</lang>
- Output:
-0.01 1.60 -1.61 1.25 -0.49 0.07