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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 = | .98+.05i  .99+.83i .94+.95i  .58+.82i  .49+.89i  .21+.89i .25+.29i  
      | .74+.88i  .4+.37i  .15+.11i  .95+.67i  .56+.31i  .43+.75i .49+.036i 
      | .67+.94i  .12+.57i .48+.098i .063+.34i .04+.89i  .9+.97i  .67+.91i  
      | .52+.38i  .69+.94i .71+.95i  .58+.98i  .53+.24i  .68+.32i .96+.47i  
      | .19+.095i .77+.16i .08+.82i  .1+.78i   .58+.39i  .93+.43i .77+.6i   
      | .78+.06i  .41+.77i .6+.64i   .09+.94i  .18+.35i  .49+.43i .36+.27i  
      | .19+.75i  .48+.16i .42+.08i  .43+.082i .24+.95i  .3+.37i  .73+.19i  
      | .94+.36i  .59+.1i  .33+.53i  .28+.22i  .53+.91i  .95+.95i .52+.73i  
      | .085+.25i .51+.57i .31+.86i  .95+.57i  .84+.94i  .51+.87i .057+.092i
      | .012+.22i .02+.51i .78+.93i  .84+.84i  .46+.053i .41+.53i .35+.063i 
      -----------------------------------------------------------------------
      .98+.9i   .68+.66i .62+.84i   |
      .92+.82i  .49+.85i .58+.76i   |
      .07+.82i  .49+.83i .92+.86i   |
      .098+.46i .44+.71i .66+.87i   |
      .69+.86i  .79+.12i .62+.54i   |
      .64+.15i  .46+.19i .055+.17i  |
      .11+.33i  .08+.35i .66+.94i   |
      .14+.14i  .99+.33i .058+.062i |
      .58+.44i  .11+.63i .97+.72i   |
      .66+.71i  .97+.67i .83+.59i   |

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

o14 = | .53+.76i  .74+.38i  |
      | .64+.16i  .49+.51i  |
      | .74+.69i  .42+.55i  |
      | .08+.78i  .6+.84i   |
      | .057+.49i .07+.83i  |
      | 1+.57i    .14+.12i  |
      | .099+.15i .11+.024i |
      | .21+.75i  .26+.25i  |
      | .95+.83i  .1+.96i   |
      | .53+.2i   .23+.16i  |

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

o15 = | -.78+.15i .43+.72i   |
      | .3-.18i   -.31+.14i  |
      | 1+.23i    .38-.45i   |
      | -.85+.16i .8+.95i    |
      | -.83-.43i -.32+.21i  |
      | 3.1-.13i  -.31-.058i |
      | -.71+2.3i 1.3-.84i   |
      | .75-.23i  .57-.69i   |
      | -.71-1.8i -1.1-.73i  |
      | -1.1+.06i -.21+.81i  |

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

o16 = 1.99840144432528e-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 = | .42 .97 .27  .19 .41  |
      | .92 .98 .65  .19 1    |
      | .77 .31 .048 .84 .38  |
      | .43 .96 .88  .23 .088 |
      | .45 .87 .78  .55 .61  |

                 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 = | -.71  1.1  .94  1.2  -2   |
      | 1.8   -.58 -.2  .025 -.14 |
      | -1.6  .32  -.36 .81  .67  |
      | .12   -1   .67  -.42 1.3  |
      | -.071 .57  -.56 -1.5 1.3  |

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

o20 = 3.33066907387547e-16

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

o21 = 9.99200722162641e-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 = | -.71  1.1  .94  1.2  -2   |
      | 1.8   -.58 -.2  .025 -.14 |
      | -1.6  .32  -.36 .81  .67  |
      | .12   -1   .67  -.42 1.3  |
      | -.071 .57  -.56 -1.5 1.3  |

                 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 :