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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 = | .19+.34i .5+.66i  .7+.84i   .32+.084i .66+.4i  .25+.93i .48+.64i
      | .93+.04i .67+.87i .41+.92i  .94+.37i  .85+.14i .52+.43i .89+.33i
      | .13+.95i .72+.31i .59+.73i  .37+.14i  .91+.12i .66+.21i .27+.34i
      | .6+.02i  .85+.3i  .69+.98i  .43+.48i  .67+.57i .41+.84i .19+.83i
      | .82+.73i .48+.19i .23+.34i  .58+.05i  .1+.92i  .47+.71i .68+.86i
      | .39+.22i .4+.78i  .67+.93i  .59+.01i  .45+.48i .06+.44i .98+.41i
      | .86+.2i  .77+.06i .11+.17i  .47i      .21+.66i .89+.1i  .66+.8i 
      | .28+.04i .84+.23i .89+.97i  .93+.68i  .24+.22i .13+.21i .75+.72i
      | .87+.2i  .91+.79i .12+.66i  .45+.21i  .41+.27i .37+.16i .63+.57i
      | .14+.52i .54+.74i .11+.025i .5+.9i    .41+.9i  .78+.63i .41+.58i
      -----------------------------------------------------------------------
      .83+.22i    .96+.48i  .45+.07i |
      .16+.24i    .91+.97i  .04+.82i |
      .1+.23i     .98+.65i  .005+.3i |
      .98+.69i    .45+.087i .96+.51i |
      .14+.31i    .21+.6i   .91+.64i |
      .049+.0014i .81+.33i  .92+.2i  |
      .088+.15i   .83+.43i  .19+.93i |
      .99+.6i     .12+.045i .29+.64i |
      .43+.92i    .17+.97i  .42+.57i |
      .97+.74i    .21+.33i  .3+.95i  |

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

o14 = | .77+.87i .4+.48i    |
      | .89+.89i .27+.34i   |
      | .54+.02i .7+.4i     |
      | .47+.97i .36+.95i   |
      | .53+.81i .16+.35i   |
      | .66+.21i .87+.29i   |
      | .82+.96i .53+.35i   |
      | .99+.41i .069+.048i |
      | .24+.23i .31+.59i   |
      | .2+.41i  .65+.24i   |

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

o15 = | -.9+.44i  .045-.22i |
      | .03+.61i  -.22-.57i |
      | -.68-.48i .94+.46i  |
      | -.07+1.4i -.16-1.7i |
      | .99+.2i   -.77+.34i |
      | .3-.64i   .51+.68i  |
      | 1.6-i     -.57+.35i |
      | -.18-.35i -.17+.24i |
      | -.19+.86i .61-.87i  |
      | -.79-.78i 1.3+.73i  |

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

o16 = 1.21745459266472e-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 = | .98  .17  .34   .79 .45  |
      | .53  .078 .0097 .46 .48  |
      | .033 .049 .12   .25 .55  |
      | .68  .6   .32   .16 .85  |
      | .089 .11  .23   .7  .025 |

                 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 = | 1.1  .25  -.94 -.075 -1   |
      | -2.4 1.7  -2.7 2     2.1  |
      | 3    -5.2 2.5  -.23  -.8  |
      | -.74 1.4  -.36 -.22  1.5  |
      | -.15 .3   1.8  -.025 -.64 |

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

o20 = 2.63677968348475e-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 = | 1.1  .25  -.94 -.075 -1   |
      | -2.4 1.7  -2.7 2     2.1  |
      | 3    -5.2 2.5  -.23  -.8  |
      | -.74 1.4  -.36 -.22  1.5  |
      | -.15 .3   1.8  -.025 -.64 |

                 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 :