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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 = | .19+.87i .93+.71i  .02+.52i .1+.92i   .2+.91i   .14+.45i  .47+.39i 
      | .45+.33i .36+.66i  .97+.13i .64+.15i  .32+.057i .61+.55i  .1+.073i 
      | .16+.93i .11+.18i  .9+.23i  .29+.12i  .85+.64i  .57+.45i  .23+.84i 
      | .91+.91i .36+.043i .61+.12i .71+.07i  .52+.47i  .71+.62i  .88+.44i 
      | .4+.26i  .58+.08i  .28+.53i .36+.092i .82+.66i  .14+.22i  .84+.95i 
      | .93+.62i .78+.23i  .04+.54i .58+.9i   .17+.68i  .69+.08i  .32+.21i 
      | .82+.16i .89+.71i  .97+.35i .74+.04i  .41+.2i   .46+.34i  .2+.58i  
      | .89+.76i .53+.19i  .89+.87i .1+.001i  .49+.7i   .76+.82i  .83+.86i 
      | .74+.73i .83+.37i  .87+.53i .15+.31i  .82+.65i  .94+.29i  .48+.074i
      | .69+.76i .17+.49i  .51+.97i .39+.93i  .98+.68i  .16+.075i .086+.32i
      -----------------------------------------------------------------------
      .23+.44i   .35+.81i   .76+i    |
      .83+.16i   .78+.23i   .82+.82i |
      .042+.011i .71+.81i   .66+.06i |
      .22+.4i    .009+.064i .32+.35i |
      .17+.095i  .72+.51i   .14+.78i |
      .8+.92i    .41+.57i   .51+.75i |
      .67+.13i   .06+.51i   .15+.46i |
      .87+.15i   .48+.52i   .55+.92i |
      .77+.34i   .87+.67i   .83+.12i |
      .79+.91i   .15+.87i   .41+.44i |

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

o14 = | .35+.84i .2+.87i  |
      | .6+.44i  .98+.62i |
      | .56+.47i .5+.21i  |
      | .47+.75i .91+.05i |
      | .28+.83i .52+.86i |
      | .87+.54i .64+.42i |
      | .36+.81i .29+.13i |
      | .96+.68i .51+.18i |
      | .62+.88i .41+.7i  |
      | .61+.41i .1+.43i  |

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

o15 = | .74-.97i   -1-.84i    |
      | .21+.82i   .27+.25i   |
      | .42-.41i   -.88-.37i  |
      | -.24+.41i  .85-.06i   |
      | -.2+.93i   1.2+.98i   |
      | 1.1i       1.7+.17i   |
      | -.54-.06i  -.36+.054i |
      | -.48+.036i .19+.75i   |
      | -.036-.27i -.33+.59i  |
      | .6-.99i    -.18-.98i  |

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

o16 = 7.7715611723761e-16

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 = | .077 .2   .23  .98 .96 |
      | .37  .44  .35  .93 .95 |
      | .17  .2   .051 .32 .34 |
      | .95  .71  .87  .27 .68 |
      | .024 .023 .22  .82 .19 |

                 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 = | 15   -34  38  6.5   3.8  |
      | -21  41   -38 -7.6  -4   |
      | -3.5 10   -16 -.83  -.59 |
      | -.15 -1.1 2.8 -.075 1.6  |
      | 5.3  -7.3 6.1 1.4   -.93 |

                 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 = 3.5527136788005e-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 = | 15   -34  38  6.5   3.8  |
      | -21  41   -38 -7.6  -4   |
      | -3.5 10   -16 -.83  -.59 |
      | -.15 -1.1 2.8 -.075 1.6  |
      | 5.3  -7.3 6.1 1.4   -.93 |

                 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 :