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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 = | .35+.22i .63+.4i   .21+.42i .055+.28i  .98+.59i .91+.42i .96+.69i 
      | .63+.22i .06+.95i  .1+.39i  .042+.063i .36+.61i .39+.89i .73+.2i  
      | .42+.44i .26+.074i .12+.2i  .5+.89i    .53+.3i  .3+.27i  .39+.66i 
      | .28+.15i .33+.73i  .51+.45i .96+.74i   .03+.86i .57+.8i  .99+.84i 
      | .031+.3i .89+.14i  .04+.71i .97+.87i   .29+.81i .06+.91i .54+.54i 
      | .52+.94i .4+.3i    1+.37i   .73+.45i   .29+.09i 1+.1i    .089+.14i
      | .07+.78i .31+.96i  .41+.06i .53+.49i   .36+.8i  .56+.65i .79+.21i 
      | .23+.76i .01+.94i  .51+.13i .57+.47i   .91+.41i .13+.8i  .06+.99i 
      | .01+.78i .6+.22i   .56+.34i .62+.36i   .86+.16i .72+.08i .74+.13i 
      | .42+.13i .6+.19i   .83+.19i .67+.73i   .74+.36i .4+.36i  .69+.17i 
      -----------------------------------------------------------------------
      .61+.97i .68+.42i  .67+.08i  |
      .82+.72i .95+.22i  .04+.31i  |
      .77+.38i .97+.98i  .19+.048i |
      .23+.94i .092+.48i .05+.52i  |
      .83+.3i  .027+.22i .87+.39i  |
      .39+.6i  .5+.64i   .14+.24i  |
      .76+.16i .32+.44i  .99+.44i  |
      .88+.86i .73+.2i   .92+.97i  |
      .9+.67i  .51+.28i  .74+.96i  |
      .54+.91i .7+.83i   .92+.61i  |

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

o14 = | .62+.17i  .67+.54i  |
      | .16+.69i  .8+.29i   |
      | .31+.7i   .9+.24i   |
      | .74+.02i  .77+.69i  |
      | .66+.5i   .3+.52i   |
      | .67+.86i  .75+.65i  |
      | .37+.061i .64+.18i  |
      | .63+.23i  .028+.29i |
      | .38+.44i  .46+.024i |
      | .06+.52i  .72+.66i  |

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

o15 = | .5+.24i   .044-.33i |
      | -.85+.31i .22+.71i  |
      | -.22-.38i .49+.91i  |
      | .43-.34i  -.34-.33i |
      | -.22-.72i -.32-.49i |
      | 1+.58i    .26-.54i  |
      | -.46-.39i .44+.1i   |
      | .34-.094i .27-.089i |
      | -.11+.73i .4+.12i   |
      | .43+.003i -.49+.1i  |

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

o16 = 4.96506830649455e-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 = | .92 .77 .012 .27 .074 |
      | .2  .89 .02  .25 .86  |
      | .59 .61 .6   .45 .29  |
      | .75 .49 .13  .36 .16  |
      | .98 .5  .41  .74 .23  |

                 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 = | -4.2 -.92 .52  12  -4.5 |
      | 5.2  .71  .087 -11 3.1  |
      | -2.1 -.78 2.6  4   -2.5 |
      | 5    1.1  -2.3 -15 8    |
      | -5.8 .35  .4   13  -4.4 |

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

o20 = 1.77635683940025e-15

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

o21 = 1.33226762955019e-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 = | -4.2 -.92 .52  12  -4.5 |
      | 5.2  .71  .087 -11 3.1  |
      | -2.1 -.78 2.6  4   -2.5 |
      | 5    1.1  -2.3 -15 8    |
      | -5.8 .35  .4   13  -4.4 |

                 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 :