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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 = | .58+.81i  .61+.94i  .11+.16i  1+.68i    .32+.91i .08+.72i  .11+.73i 
      | .033+.32i .016+.22i .81+.69i  .82+.78i  .13+.17i .7+.85i   .86+.88i 
      | .18+.18i  .62+.68i  .86+.95i  .71+.63i  .44+.19i .6+.5i    .65+.32i 
      | .82+.67i  .27+.14i  .86+.42i  .17+.53i  .47+.66i .41+.72i  .4+.93i  
      | .63+.03i  .62+.41i  .01+.043i .95+.93i  .2+.7i   .081+.43i .73+.23i 
      | .16+.81i  .25+.11i  .21+.98i  .04+.95i  .34+.75i .89+.07i  .62+.83i 
      | .52+.01i  .05+.51i  .92+.22i  .21+.26i  .65+.29i .38+.43i  .65+.44i 
      | .86+.95i  .19+.001i .65+.04i  .72+.34i  .44+.86i .93+.14i  .64+.22i 
      | .72+.26i  .99+.1i   .31+.3i   .004+.28i .69+.42i .38+.31i  .13+.058i
      | .86+.85i  .9+.59i   1+.81i    .21+.94i  .26+.5i  .11+.75i  .44+.12i 
      -----------------------------------------------------------------------
      .95+.05i .097+.19i .9+.45i  |
      .92+.78i .51+.42i  .98+.93i |
      .34+.91i .24+.044i .02+.19i |
      .6+.15i  .36+.74i  .6+.81i  |
      .8+.71i  .98+.57i  .36+.92i |
      .7+.18i  .69+.41i  .74+.71i |
      .13+.45i .29+.16i  .56+i    |
      .69+.45i .71+.56i  .27+.19i |
      .9+.3i   .82+.99i  .95+.84i |
      .81+.57i .13+.31i  .52+.61i |

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

o14 = | .64+.55i .56+.56i |
      | .57+.83i .74+.04i |
      | .66+.3i  .92+.84i |
      | .16+.59i .63+.57i |
      | .43+.78i .27+.77i |
      | .65+.83i .27+.19i |
      | .06+.77i .44+.73i |
      | .42+.15i .84+.32i |
      | .25+.81i .7+.03i  |
      | .82+.68i .98+.47i |

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

o15 = | .12+.25i   -1-i       |
      | .28+.25i   -.37+.083i |
      | .015-.17i  -.27+.7i   |
      | -.078-.27i -.2-.87i   |
      | -.14-.44i  2.6+1.1i   |
      | .045+.1i   .12-1.5i   |
      | .23-.002i  .56+.85i   |
      | .28-.33i   1.1+.5i    |
      | -.27+.44i  -.27-.2i   |
      | .45+.45i   -1.1+.06i  |

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

o16 = 8.08254562088053e-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 = | .49 .97   .49 .27 .55 |
      | .46 .0013 .91 .92 .12 |
      | .76 .56   .64 .67 .64 |
      | .76 .17   .28 .41 .49 |
      | .97 .2    .73 .45 .16 |

                 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 = | 7.9 5.5 -17 13  -2.4 |
      | 16  11  -31 22  -6.5 |
      | -24 -18 50  -38 11   |
      | 23  19  -48 36  -11  |
      | -24 -18 48  -34 8.9  |

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

o20 = 7.105427357601e-15

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

o21 = 5.32907051820075e-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 = | 7.9 5.5 -17 13  -2.4 |
      | 16  11  -31 22  -6.5 |
      | -24 -18 50  -38 11   |
      | 23  19  -48 36  -11  |
      | -24 -18 48  -34 8.9  |

                 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 :