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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 = | 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 = | .34+.96i .36+.53i .63+.21i .45+.27i .69+.58i  .84+.24i .73+.74i
      | .55+.39i .6+.96i  .66+.99i .3+.067i .14+.54i  .47+.19i .09+.56i
      | .11+.55i .34+.59i .18+.95i .23+.29i .12+.18i  .32+.76i .56+.84i
      | .77+.62i .16+.5i  .36+.28i .39+.27i .44+.96i  .95+.54i .12+.7i 
      | .64+.96i .35+.49i .58+.17i .16+.72i .55+.21i  .9+.79i  .41+.67i
      | .02+.89i .9+.98i  .43+.97i .59+.25i .34+.95i  .72+.94i .43+.66i
      | .61+.8i  .65+.61i .16+.85i .46+.68i .69+.4i   .03+.56i .82+.45i
      | .4+.72i  .71+.89i .14+.83i .07+.94i .071+.25i .81+.06i .47+.73i
      | .4+.46i  .97+.76i .88+.42i .49+.75i .03+.7i   .73+.89i .99+.62i
      | .44+.56i .59+.61i .31+.69i .062+.2i .92+.71i  .87+.52i .44+.79i
      -----------------------------------------------------------------------
      .3+.22i  .66+.42i  .1+.98i  |
      .39+.47i .2+.74i   .24+.32i |
      .1+.98i  .48+.38i  .26+.41i |
      .92+.35i .18+.6i   .64+.83i |
      .37+.88i .27+.053i .56+.12i |
      .96+.57i .48+.41i  .12+.27i |
      .67+.72i .63+.95i  .6+.14i  |
      .32+.23i .39+.12i  .46+.7i  |
      .35+.89i .19+.48i  .53+.06i |
      .7+.04i  .64+.58i  .76+.04i |

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

o14 = | .25+.87i  .36+.57i  |
      | .15+.77i  .082+.32i |
      | .07+.79i  .78+.24i  |
      | .28+.74i  .91+.29i  |
      | .41+.68i  .52+.57i  |
      | .45+.028i .2+.73i   |
      | .76+.27i  .3+.6i    |
      | .14+.56i  .55+.52i  |
      | .43+.51i  .68+.15i  |
      | .51+.83i  .07+.18i  |

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

o15 = | 1.3+1.1i  -.5+1.6i   |
      | -.49-.61i .41+.4i    |
      | -.23-.27i .17-.48i   |
      | -.82-.5i  .54-.53i   |
      | 1.2-2i    1.3+1.2i   |
      | -.87+.51i .22-.86i   |
      | .85+.16i  .025-.036i |
      | .27-.92i  .43-.26i   |
      | .37+1.8i  -1.4+.15i  |
      | -1.2+.26i .26-1.7i   |

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

o16 = 9.42055475210265e-16

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 = | .23 .062 .49 .97 .77  |
      | .12 .6   .94 .6  .4   |
      | .19 .5   .15 .61 .052 |
      | .56 .41  .99 .93 .031 |
      | .93 .59  .42 .27 .87  |

                 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 = | -.25 -1.2 -.22 .97  .76  |
      | -.77 .86  1.4  -.74 .23  |
      | -.22 .83  -1.3 .57  -.13 |
      | .71  -.56 .89  .24  -.43 |
      | .68  .47  -.36 -.89 .38  |

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

o20 = 2.4980018054066e-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 = | -.25 -1.2 -.22 .97  .76  |
      | -.77 .86  1.4  -.74 .23  |
      | -.22 .83  -1.3 .57  -.13 |
      | .71  -.56 .89  .24  -.43 |
      | .68  .47  -.36 -.89 .38  |

                 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 :