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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 = | .4+.31i  .46+.44i .64+.11i .25+.69i  .35+.41i .91+.89i .04+.74i
      | .77+.42i .52+.13i .88+.17i .97+.72i  .51+.8i  .22+.52i .29+.96i
      | .35+.8i  .47+.73i .89+.37i .038+.19i .61+.29i .27+.4i  .15+.9i 
      | .79+.81i .32+.19i .14+.12i .87+.95i  .97+.02i .1+.58i  .7+.54i 
      | .91+.54i .18+.21i .61+.7i  .31+.5i   .18+.38i .86+.2i  .85+.01i
      | .35+.65i .72+.04i .48+.9i  .36+.95i  .75+.51i .77+.02i .48+.63i
      | .18+.11i .055+.4i .92+.29i .51+.72i  .95+.48i .99+.86i .17+.87i
      | .98+.12i .95+.48i .89+.98i .48+.054i .99+.98i .29+.26i .72+.86i
      | .91+.87i .96+.08i .05+.98i .56+.09i  .92+.74i .98+.99i .12+.33i
      | .97+.08i .15+.74i .65+.25i .55+.34i  .58+.72i .24+.31i .04+.9i 
      -----------------------------------------------------------------------
      .47+.16i  .51+.02i .77+.79i  |
      .29+.008i .58+.32i .018+.14i |
      .35+.49i  .85+.2i  .66+.54i  |
      .02+i     .56+.58i .85+.48i  |
      .64+.62i  .51+.9i  .32+.76i  |
      .72+.75i  .8+.47i  .37+.22i  |
      .16+.32i  .77+.53i .5+.61i   |
      .92+.14i  .49+.63i .53+.47i  |
      .16+.78i  .99+.71i .23+.36i  |
      .58+.72i  .78+.26i .077+.15i |

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

o14 = | .28+.32i .3+.051i |
      | .77+.2i  .72+.65i |
      | .67+.54i .63+.26i |
      | .4+.073i .84i     |
      | .6+.4i   .49+.13i |
      | .3+.94i  .7+.65i  |
      | .92+.32i .8+.37i  |
      | .3+.43i  .89+.15i |
      | .62+.89i .93+.65i |
      | .18+.66i .9+.78i  |

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

o15 = | 2.5-.3i    .36-.091i |
      | -3.5-1.9i  -1.2-.68i |
      | -4.6-3.2i  -1-1.2i   |
      | -1.8-1.9i  -.31-.19i |
      | -.03+1.3i  .16-.21i  |
      | .019+.072i .14-.34i  |
      | 3.3-2.1i   1.3-.43i  |
      | -1.8+6.3i  -.17+1.4i |
      | 5.5+1.5i   1.7+i     |
      | -.25+.16i  -.42+.38i |

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

o16 = 2.59184174787402e-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 = | .74 .88 .82 .3  .95 |
      | .5  .88 .21 .27 .54 |
      | .35 .11 .91 .35 .46 |
      | .42 .9  .53 .32 .72 |
      | .73 .11 .31 .52 .21 |

                 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 = | 1.3  4.4  1.3  -5.8 -.35 |
      | -2.2 8.6  4    -5.3 -2.5 |
      | -1.4 7.9  5.4  -6.7 -2.7 |
      | -2.2 -6.2 -2.4 8.1  2.9  |
      | 4    -16  -8.7 13   4    |

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

o20 = 2.22044604925031e-15

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

o21 = 3.99680288865056e-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 = | 1.3  4.4  1.3  -5.8 -.35 |
      | -2.2 8.6  4    -5.3 -2.5 |
      | -1.4 7.9  5.4  -6.7 -2.7 |
      | -2.2 -6.2 -2.4 8.1  2.9  |
      | 4    -16  -8.7 13   4    |

                 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 :