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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 = | .85+.79i .91+.14i .61+.96i .97+.32i .75+.82i .53+.96i .4+.48i   
      | .97+.65i .75+.13i .77+.98i .22+.22i .94+.42i .67+.61i .48+.49i  
      | .77+.05i .9+.28i  .57+.15i .78+.33i .57+.84i .56+.46i .088+.26i 
      | .98+.23i .85+.2i  .98+.19i .99+.72i .38+.33i .65+.84i .18+.76i  
      | .062+.2i .33+.93i .8+.85i  .44+.19i .68+.73i .31+.67i .61+.33i  
      | .79+.7i  .47+.93i .34+.85i .48+.68i .85+.26i .97+.87i .43+.18i  
      | .11+.33i .63+.81i .42+.54i .98+.68i .87+.8i  .82+.88i .62+.12i  
      | .7+.76i  .24+.75i .82+.73i .4+.55i  .21+i    .21+.81i .27+.079i 
      | .31+.55i .92+.4i  .19+.19i .4+.08i  .72+.98i .91+.87i .059+.011i
      | .76+.03i .43+.92i .54+.9i  .94+.22i .6+.82i  .16+.58i .006+.09i 
      -----------------------------------------------------------------------
      .46+.64i .8+.88i   .71+.07i  |
      .17+.68i .25+.82i  .36+.13i  |
      .64+.97i .24+.017i .64+.5i   |
      .65+.88i .02+.6i   .16+.69i  |
      .3+.044i .65+.78i  .96+.03i  |
      .06+.73i .35+.94i  .46+.052i |
      .91+.28i .92+.64i  .34+.77i  |
      .73+.12i .26+.048i .015+.13i |
      .24+.64i .091+.39i .18+.38i  |
      .08+.9i  .71+.68i  .32+.8i   |

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

o14 = | .39+.75i  .75+.93i |
      | .83+.24i  .1+.12i  |
      | .55+.09i  .32+.44i |
      | .072+.21i .29+.63i |
      | .25+.21i  .84+.02i |
      | .47+.19i  .96+.59i |
      | .24+.76i  .78+.48i |
      | .04+.61i  .2+.36i  |
      | .23+.053i .9+.32i  |
      | .97+.5i   .98+.68i |

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

o15 = | .57+.51i   -.39+.33i  |
      | .37+.028i  -.81-.15i  |
      | -.36-.65i  .24-.25i   |
      | -.017+.36i .6+.53i    |
      | .39+.28i   -.074-.15i |
      | -.43-.41i  1.2-.46i   |
      | 1.1+.43i   -1.3+.08i  |
      | -.45-.17i  .05-.11i   |
      | -.2+.66i   .27-.37i   |
      | .08-.53i   .098+.5i   |

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

o16 = 8.95090418262362e-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 = | .38  .048 .014 .49  .18 |
      | .4   .2   .79  .079 .3  |
      | .025 .054 .37  .99  .11 |
      | .89  .77  .41  .77  .67 |
      | .45  .56  .2   .34  .97 |

                 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 = | 2.3  .53  -1.2   .26  -.63  |
      | -3.3 -1   .24    2.2  -.58  |
      | -.73 1.2  .45    -.16 -.18  |
      | .29  -.44 .86    .093 -.083 |
      | .93  .27  -.0061 -1.4 1.7   |

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

o20 = 1.11022302462516e-16

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

o21 = 2.22044604925031e-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 = | 2.3  .53  -1.2   .26  -.63  |
      | -3.3 -1   .24    2.2  -.58  |
      | -.73 1.2  .45    -.16 -.18  |
      | .29  -.44 .86    .093 -.083 |
      | .93  .27  -.0061 -1.4 1.7   |

                 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 :