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solve -- solve a linear equation

Synopsis

Description

(Disambiguation: for division of matrices, which can also be thought of as solving a system of linear equations, see instead Matrix // Matrix. For lifting a map between modules to a map between their free resolutions, see extend.)

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 = | 2.2e-16  |
      | -2.2e-16 |
      | 0        |

                 3          1
o10 : Matrix RR    <--- RR
               53         53
i11 : norm oo

o11 = 2.22044604925031e-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 = | .21+.4i    .33+.71i .87+.14i  .44+.2i  .9+.49i   .89+.26i .74+.15i
      | .088+.008i .7+.75i  .56+.85i  .2+.97i  .15+.12i  .72+.43i .43+.16i
      | .33+.44i   .24+.92i .37+.16i  .96+.03i .84+.71i  .23+.44i .78+.93i
      | .003+.2i   .43+.47i .15+.27i  .17+.69i .62+.99i  .64+.19i .03+.64i
      | .16+.7i    .73+.71i .21+.36i  .19+.78i .97+.04i  .57+.75i .08+.68i
      | .74+.37i   .74+.73i .24+.56i  .15+.85i .54+.38i  .38+.88i .53+.35i
      | .42+.23i   .13+.94i .37+.97i  .84+.92i .68+.3i   .65+.76i .39+.12i
      | .23+.7i    .21i     .018+.41i .58+.57i .37+.079i .08+.96i .89+.01i
      | .12+.73i   .07+.53i .4+.72i   .72+.38i .62+.81i  .43+.44i .94+.33i
      | .8+.82i    .64+.36i .95+.81i  .66+.43i .75+.27i  .87+.5i  .29+.65i
      -----------------------------------------------------------------------
      .031+.44i .24+.76i .58+.66i  |
      .8+.6i    .47+.38i .77+.12i  |
      .07+.97i  .19+.38i .46+.58i  |
      .09+.78i  .73+.98i .055+.44i |
      .25+.66i  .42+.14i .47+.45i  |
      .63+.74i  .9+.4i   .34+.33i  |
      .48+.1i   .13+.18i .15+.032i |
      .63+.72i  .19+.47i .27+.61i  |
      .65+.12i  .71+.97i .19+.46i  |
      .39+.024i .77+.64i .79+.55i  |

                 10          10
o13 : Matrix CC     <--- CC
               53          53
i14 : b = random(CC^n,CC^2)

o14 = | .24+.26i   .46+.52i  |
      | .28+.78i   .081+.33i |
      | .87+.71i   .89+.63i  |
      | .14+.072i  .63+.03i  |
      | .81+.69i   .58+.87i  |
      | .69+.94i   .92+.12i  |
      | .59+.24i   .39+.94i  |
      | .47+.85i   .49+.95i  |
      | .63+.52i   .16+.45i  |
      | .011+.096i .75+.89i  |

                 10          2
o14 : Matrix CC     <--- CC
               53          53
i15 : x = solve(A,b)

o15 = | -.51-.44i .77-.61i   |
      | 1.1-.65i  -.39-.043i |
      | -1.7+.7i  .62+.31i   |
      | .71+.09i  .08+.71i   |
      | -.79-.64i -.2+.23i   |
      | .9-.72i   .87-.28i   |
      | .94+.05i  -.47-.15i  |
      | -.68+.91i .44-1.3i   |
      | .36+.69i  .2+.058i   |
      | .55+.14i  -1+.67i    |

                 10          2
o15 : Matrix CC     <--- CC
               53          53
i16 : norm ( matrix A * matrix x - matrix b )

o16 = 1.15910686703364e-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 = | .71 .74 .87  .48  .84 |
      | .2  .66 .72  .31  .35 |
      | .28 .11 .013 .64  .25 |
      | .15 .21 .47  .037 .14 |
      | .97 .65 .076 .37  .13 |

                 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 = | .21  -1.8 .11  2.3  .86  |
      | -.17 2.6  -1.2 -3.7 .53  |
      | -.62 -.45 .61  4    -.26 |
      | -1   .61  2    .89  -.11 |
      | 2.4  -.6  -.79 -3.4 -.87 |

                 5          5
o19 : Matrix RR    <--- RR
               53         53
i20 : norm(A*A' - I)

o20 = 4.9960036108132e-16

o20 : RR (of precision 53)
i21 : norm(A'*A - I)

o21 = 6.66133814775094e-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 = | .21  -1.8 .11  2.3  .86  |
      | -.17 2.6  -1.2 -3.7 .53  |
      | -.62 -.45 .61  4    -.26 |
      | -1   .61  2    .89  -.11 |
      | 2.4  -.6  -.79 -3.4 -.87 |

                 5          5
o22 : Matrix RR    <--- RR
               53         53
i23 : norm(A' - A'')

o23 = 0

o23 : RR (of precision 53)

Caveat

This function is limited in scope, but is sometimes useful for very large matrices

See also

Ways to use solve :