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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 = | .39+.95i  .7+.05i  .08+.72i .14+.38i   .93+.22i .13+.44i .23+.77i
      | .13+.43i  .72+.75i .03+.77i .07+.25i   .72+.64i .53+.96i .07+.73i
      | .081+.33i .9+.47i  .98+.28i .1+.88i    .96+.91i .41+.24i .4+.32i 
      | .051+.12i .5+.66i  .81+.14i .033+.1i   .29+.42i .27+.37i .09+.55i
      | .19+.3i   .58+.72i .61+.29i .036+.048i .62+.03i .04+.52i .24+.5i 
      | .99+.13i  .4+.81i  .78+.58i .53+.63i   .86+.95i .27+.37i .5+.41i 
      | .08+.98i  .6+.27i  .74+.44i .56+.34i   .35+.65i .93+.13i .17+.98i
      | .53+.91i  .35+.74i .97+.8i  .98+.15i   .33+.33i .39+.83i .64+.96i
      | .67+.91i  .94+.73i .04+.58i .9+.29i    .79+.13i .3+.89i  .18+.54i
      | .52+.92i  .61+.77i .23+.73i .7+.37i    .14+.99i .17+.3i  .67+.82i
      -----------------------------------------------------------------------
      .25+.86i .23+.37i .96+.34i     |
      .1+.065i .33+.3i  .81+.26i     |
      .86+.99i .88+.96i .71+.03i     |
      .45+.21i .83+.03i .18+.79i     |
      .82+.7i  .49+.73i .36+.96i     |
      .63+.12i .46+.34i .0072+.0039i |
      .77+.07i .16+.1i  .07+.99i     |
      .88+.63i .02+.74i .09+.91i     |
      .13+.2i  .19+.51i .75+.81i     |
      .11+.93i .16+.56i .41+.17i     |

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

o14 = | .025+.11i .98+.71i |
      | .29+.039i .89+.19i |
      | .41+.64i  .94+.05i |
      | .47+.52i  .22+.5i  |
      | .74+.15i  .05+.73i |
      | .047+.38i .71+.81i |
      | .33+.012i .14+.45i |
      | .21+.072i .15+.36i |
      | .9+.29i   .96+.34i |
      | .13+.75i  .52+.52i |

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

o15 = | -1.4-1.4i -.58+2.4i |
      | 1-4.5i    -3.3+1.5i |
      | 2.6-2.7i  -3.1-1.6i |
      | 1.4+1.9i  1.4-2.3i  |
      | -.09+.87i .75-.33i  |
      | -1.3+3.3i 2.7-.42i  |
      | -3.8+3i   3.6+1.9i  |
      | 2.3-.2i   -.6-1.4i  |
      | -2.1+2.6i 2.3+.89i  |
      | 3.1+1.6i  .99-3.2i  |

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

o16 = 3.21964677141295e-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 = | .47 .46  .86 .71 .085 |
      | .15 .54  .63 .59 .03  |
      | .56 .061 .12 .69 .44  |
      | .86 .68  .7  .65 .5   |
      | .65 .45  .23 .32 .86  |

                 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 = | -.76 -1.8 .22  3    -1.7 |
      | -3.7 2.6  -.45 2.6  -.99 |
      | 4.1  -1.7 -1.2 -2.4 1.6  |
      | -.86 1.6  1.7  -.49 -.53 |
      | 1.7  -.16 -.21 -2.8 2.7  |

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

o20 = 6.66133814775094e-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 = | -.76 -1.8 .22  3    -1.7 |
      | -3.7 2.6  -.45 2.6  -.99 |
      | 4.1  -1.7 -1.2 -2.4 1.6  |
      | -.86 1.6  1.7  -.49 -.53 |
      | 1.7  -.16 -.21 -2.8 2.7  |

                 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 :