There are several restrictions. The first is that there are only a limited number of rings for which this function is implemented. Second, over
RR or
CC, the matrix
A must be a square non-singular matrix. Third, if
A and
b are mutable matrices over
RR or
CC, they must be dense matrices.
i1 : kk = ZZ/101;
|
i2 : A = matrix"1,2,3,4;1,3,6,10;19,7,11,13" ** kk
o2 = | 1 2 3 4 |
| 1 3 6 10 |
| 19 7 11 13 |
3 4
o2 : Matrix kk <--- kk
|
i3 : b = matrix"1;1;1" ** kk
o3 = | 1 |
| 1 |
| 1 |
3 1
o3 : Matrix kk <--- kk
|
i4 : x = solve(A,b)
o4 = | 2 |
| -1 |
| 34 |
| 0 |
4 1
o4 : Matrix kk <--- kk
|
i5 : A*x-b
o5 = 0
3 1
o5 : Matrix kk <--- kk
|
Over
RR or
CC, the matrix
A must be a non-singular square matrix.
i6 : printingPrecision = 2;
|
i7 : A = matrix "1,2,3;1,3,6;19,7,11" ** RR
o7 = | 1 2 3 |
| 1 3 6 |
| 19 7 11 |
3 3
o7 : Matrix RR <--- RR
53 53
|
i8 : b = matrix "1;1;1" ** RR
o8 = | 1 |
| 1 |
| 1 |
3 1
o8 : Matrix RR <--- RR
53 53
|
i9 : x = solve(A,b)
o9 = | -.15 |
| 1.1 |
| -.38 |
3 1
o9 : Matrix RR <--- RR
53 53
|
i10 : A*x-b
o10 = | 0 |
| -3.3e-16 |
| -8.9e-16 |
3 1
o10 : Matrix RR <--- RR
53 53
|
i11 : norm oo
o11 = 8.88178419700125e-16
o11 : RR (of precision 53)
|
For large dense matrices over
RR or
CC, this function calls the lapack routines.
i12 : n = 10;
|
i13 : A = random(CC^n,CC^n)
o13 = | .35+.46i .51+.19i .13+.31i .76+.43i .86+.33i .2+.92i .83+.7i
| .081+.4i .67+.18i .44+.94i .54+.64i .43+.85i .42+.052i .42+.36i
| .12+.23i .52+.9i .88+.84i .094+.11i .12+.32i .61+.02i .94+.69i
| .96+.1i .52+.11i .88+.33i .34+.65i .99+.01i .57+.75i .026+.11i
| .56+.6i .56+.61i .25+.093i .35+.83i .56+.2i .61+.17i .63+.59i
| .21+.79i .39+.79i .38+.76i .18+.035i .81+.36i .94+.11i .57+.43i
| .97+.99i .54+.8i .97+.93i .17+.65i .64+.88i .37+.51i .81
| .21+.87i .44+.57i .07+.71i .84+.45i .31+.49i .53+.55i .71+.25i
| .44+.14i .3+.43i .54+.66i .94+.09i .29+.85i .83+.04i .61+.74i
| .64+.94i .63i .58+.04i .99+.32i .15+.12i .99i .54+.39i
-----------------------------------------------------------------------
.38+.52i .44+.97i .37+.14i |
.45+.32i .14+.67i .33+.71i |
.52+.78i .75+.63i .4+.71i |
.69+.02i .13+.24i .37+.1i |
.66+.13i .6+.81i .69+.9i |
.032+.25i .84+.6i .51+.74i |
.23+.025i .95+.67i .75+.06i |
.96+.79i .72+.69i .7+.99i |
.074+.43i .67+.65i .05+.93i |
.41+.79i .73+.15i .92+.55i |
10 10
o13 : Matrix CC <--- CC
53 53
|
i14 : b = random(CC^n,CC^2)
o14 = | .6+.37i .76+.25i |
| .48+.16i .01+.87i |
| .42+.14i .68+.76i |
| .51i .32+.32i |
| .72+.19i .29+.99i |
| .66+.94i .66+.3i |
| .25+.25i .5+.028i |
| .46+.56i .92+.13i |
| .74+.43i .79+.8i |
| .78+.99i .59+.34i |
10 2
o14 : Matrix CC <--- CC
53 53
|
i15 : x = solve(A,b)
o15 = | -.82-3i .29+6.4i |
| -4.2-6.3i 2.2+19i |
| -.34-.95i -.72+3.2i |
| 5.2-2.5i -14+.7i |
| 4.1+1.3i -7.7-7.4i |
| -.58+4.9i 7.7-8.4i |
| 4.8-1.6i -10-2.4i |
| 3.2+1.6i -5.4-8.3i |
| -4.4+4.8i 13-6.2i |
| -4.6+3.4i 14-1.9i |
10 2
o15 : Matrix CC <--- CC
53 53
|
i16 : norm ( matrix A * matrix x - matrix b )
o16 = 6.53147139932152e-15
o16 : RR (of precision 53)
|
This may be used to invert a matrix over
ZZ/p,
RR or
QQ.
i17 : A = random(RR^5, RR^5)
o17 = | .029 .59 .59 .4 .65 |
| .99 .87 .18 .99 .92 |
| .39 .54 .11 .75 .95 |
| .75 .13 .8 .36 .19 |
| .92 .36 .58 .99 .67 |
5 5
o17 : Matrix RR <--- RR
53 53
|
i18 : I = id_(target A)
o18 = | 1 0 0 0 0 |
| 0 1 0 0 0 |
| 0 0 1 0 0 |
| 0 0 0 1 0 |
| 0 0 0 0 1 |
5 5
o18 : Matrix RR <--- RR
53 53
|
i19 : A' = solve(A,I)
o19 = | -1.4 .7 1.1 1.9 -1.7 |
| 1.4 1.9 -2.4 -.88 -.39 |
| .89 -.6 -.37 .54 .33 |
| 1.1 -.26 -2.8 -3.4 4.2 |
| -1.2 -1.1 4.2 2.3 -2.4 |
5 5
o19 : Matrix RR <--- RR
53 53
|
i20 : norm(A*A' - I)
o20 = 4.44089209850063e-16
o20 : RR (of precision 53)
|
i21 : norm(A'*A - I)
o21 = 7.7715611723761e-16
o21 : RR (of precision 53)
|
Another method, which isn't generally as fast, and isn't as stable over
RR or
CC, is to lift the matrix
b along the matrix
A (see
Matrix // Matrix).
i22 : A'' = I // A
o22 = | -1.4 .7 1.1 1.9 -1.7 |
| 1.4 1.9 -2.4 -.88 -.39 |
| .89 -.6 -.37 .54 .33 |
| 1.1 -.26 -2.8 -3.4 4.2 |
| -1.2 -1.1 4.2 2.3 -2.4 |
5 5
o22 : Matrix RR <--- RR
53 53
|
i23 : norm(A' - A'')
o23 = 0
o23 : RR (of precision 53)
|
For division of matrices, which can also be thought of as solving a system of linear equations, see Matrix // Matrix.