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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 = | .65+.92i .69+.07i  .52+.74i   .35+.28i .22+.98i   .49+.84i .15+.45i 
      | .48+.9i  .14+.5i   .91+.34i   .64+.26i .84+.57i   .61+.26i .2+.15i  
      | .25+.82i .19+.7i   .72+.93i   .71+.71i .22+.083i  .62+.71i .24+.63i 
      | .51+.59i .84+.49i  .17+.57i   .59+.09i .21+.67i   .33+.21i .89+.07i 
      | .2+.28i  .85+.95i  .18+.82i   .48+.94i .32+.43i   .75+.94i .081+.17i
      | .67+.82i .44+.034i .007+.083i .88+.58i .79+.48i   .71+.16i .43+.49i 
      | .44+.82i .71+.49i  .99+.52i   .26+.8i  .81+.73i   .45+.58i .89+.61i 
      | .64+.49i .56+.24i  .66+.6i    .19+.53i .3+.7i     .65+.06i .74+.9i  
      | .85+.94i .19+.33i  .61+.17i   .45+.79i .053+.089i 1+.27i   .1+.13i  
      | .001+.3i .4+.053i  .94+.42i   .64+.13i .5+.89i    .83+.69i 1+.99i   
      -----------------------------------------------------------------------
      .57+.83i  .17+.76i .21+.04i  |
      .2+.2i    .24+.69i .59+.67i  |
      .72+.06i  .83+.64i .84+.26i  |
      .69+.47i  .53+.82i .71+.85i  |
      .35+.057i .51+.71i .56+.63i  |
      .95+.92i  .51+.49i .68+.03i  |
      1+.1i     .97+.95i .49+.17i  |
      .3+.67i   .76+i    .43+.92i  |
      .48+.72i  .43+.83i .05+.52i  |
      .32+.96i  .67+.05i .068+.12i |

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

o14 = | .91+.61i .28+.45i |
      | .2+.38i  .66+.9i  |
      | .41+.29i .93+i    |
      | .22+.4i  .031+.1i |
      | .07+.61i .46+.78i |
      | .94+.76i .42+.88i |
      | .4+.22i  .66+.74i |
      | .17+.68i .76+.06i |
      | .74+.68i .06+.85i |
      | .56+.97i .93+.73i |

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

o15 = | .43+.021i .82-.9i   |
      | .61+.07i  1.2-.92i  |
      | .71-.37i  .47-.85i  |
      | .056-.33i -.79+.17i |
      | .16+.49i  1+.78i    |
      | -.43+.17i -.6+.49i  |
      | -.46-.16i -.9-1.3i  |
      | 1.1-.7i   .58-.2i   |
      | -.62+i    .22+2.9i  |
      | -.23-.32i -.57-1.2i |

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

o16 = 1.02357505330418e-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 = | .56 1   .017 .43 .51 |
      | .79 .96 .48  .3  .66 |
      | .17 .53 .23  .71 .32 |
      | .1  .56 .42  .55 .26 |
      | .36 .66 .78  .32 .45 |

                 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 = | -8.6 13  -5   15  -14  |
      | 1.3  .12 -4.4 5.6 -1.8 |
      | -2.5 2.5 -2   4.3 -2   |
      | -3.4 4   .38  4.8 -5.1 |
      | 12   -18 14   -31 24   |

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

o20 = 3.5527136788005e-15

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

o21 = 4.44089209850063e-15

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 = | -8.6 13  -5   15  -14  |
      | 1.3  .12 -4.4 5.6 -1.8 |
      | -2.5 2.5 -2   4.3 -2   |
      | -3.4 4   .38  4.8 -5.1 |
      | 12   -18 14   -31 24   |

                 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 :