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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 = | .59+.55i  .18+.87i .19+.42i .29+.61i .08+.76i  .89+.1i  .78+.23i 
      | .11+.26i  .82+.72i .52+.43i .79+.47i .72+.83i  .93+.37i .46+.76i 
      | .67+.29i  .69+.06i .89+.08i .41+.57i .6+.58i   .19+.54i .98+.46i 
      | .08+.7i   .92+.08i .02+.84i .42+.65i .41+.15i  .65+.66i .25+.58i 
      | .32+.29i  .78+.22i .38+.17i .12+.84i .61+.73i  .82+.55i .95+.69i 
      | .64+.08i  .85+.32i .32+.29i .19+.91i .89+.7i   .75+.17i .44+.73i 
      | .42+.072i .69+.18i .7+.99i  .15+.69i .38+.055i .5+.58i  .059+.45i
      | .65+.95i  .43+.42i .03+.94i .7+.88i  .35+.56i  .63+.09i .31+.35i 
      | .7+.45i   .08+.98i .09+.85i .67+.84i .46+.38i  .73+.53i .09+.57i 
      | .56+.08i  .94+.02i .38+.18i .2+.56i  .95+.13i  .65+.64i .16+.62i 
      -----------------------------------------------------------------------
      .25+.34i  .99+.48i .56+.7i    |
      .52+.97i  .38+.21i .26+.096i  |
      .15+.93i  .55+.08i .34+.94i   |
      .36+.006i .53+.19i .03+.99i   |
      .84+.42i  .81+.12i .83+.7i    |
      .57+.46i  .78+.94i .85+.02i   |
      .35+.2i   .22+.47i 1+.05i     |
      .17+.06i  .49+.3i  .12+.87i   |
      .53+.72i  .7+.6i   .61+.36i   |
      .51+.71i  .5+.49i  .05+.0023i |

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

o14 = | .95+.75i  .49+.42i |
      | .061+.44i .1+.74i  |
      | .45+.087i .36+.36i |
      | .74+.4i   .11+.54i |
      | .54+.12i  .35+.88i |
      | .98+.07i  .62+.2i  |
      | .25+.34i  .81+.6i  |
      | .21+i     .4+.93i  |
      | .71+.2i   .75+.48i |
      | .5+.99i   .67+.7i  |

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

o15 = | -1.7+.51i -.3+.52i  |
      | -.69-2.8i -1.3-1.7i |
      | 2.2+2.4i  2.1+.73i  |
      | -.97-1.5i -1-i      |
      | 3.2+2.7i  2.6+1.2i  |
      | 1.6+1.8i  1.2+.57i  |
      | -2-.67i   -.86-.05i |
      | -2+1.1i   -.55+1.3i |
      | .44+.1i   -.42+.19i |
      | .67-2.9i  -.33-1.3i |

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

o16 = 2.86568757593482e-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 = | .96  .23 .18 .25 .63 |
      | .023 .92 .67 .5  .75 |
      | .53  .17 .49 .82 .87 |
      | .47  .46 .77 .82 .79 |
      | .74  .53 1   .8  .7  |

                 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 = | 6   -7.7 -20 39   -16 |
      | 15  -19  -55 110  -47 |
      | -17 23   63  -130 57  |
      | 20  -30  -80 160  -70 |
      | -16 24   64  -130 53  |

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

o20 = 1.77635683940025e-14

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

o21 = 2.66453525910038e-14

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 = | 6   -7.7 -20 39   -16 |
      | 15  -19  -55 110  -47 |
      | -17 23   63  -130 57  |
      | 20  -30  -80 160  -70 |
      | -16 24   64  -130 53  |

                 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 :