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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 = | -1.1e-16 |
      | -7.8e-16 |
      | 0        |

                 3          1
o10 : Matrix RR    <--- RR
               53         53
i11 : norm oo

o11 = 7.7715611723761e-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 = | .17+.29i .55+.81i .5+.28i  .65+.03i .21+.63i  .85+.42i .44+.36i 
      | .83+.87i .87+.54i .79+.41i .27+.53i .23+.98i  .38+.7i  .087+.37i
      | .59+.17i .5+.56i  .68+.79i .95+.33i .62+.7i   .68+.99i .91+.9i  
      | .54+.58i .4+.22i  .9+.51i  .21+.73i .041+.13i .63+.82i .38+.21i 
      | .44+.43i .36+.73i .72+.95i .44+.93i .31+.34i  .29+.58i .11+.42i 
      | .76+.77i 1+.38i   .37+.58i .53+.87i .29+.12i  .75+.5i  .64+.57i 
      | .82+.37i .78+.89i .55+.99i .8+.3i   .41+.25i  .92+.18i .89+.33i 
      | .89+.53i .28+.49i .78+.85i .08+.5i  .03+.65i  .53+.28i .65+.96i 
      | .63+.78i .11+.97i .34+.47i .33+.48i .8+.67i   .58+.47i .24+.68i 
      | .89+.23i .27+.46i .13+.46i .38+.5i  .19+.82i  .02+.51i .9+.06i  
      -----------------------------------------------------------------------
      .14+.82i  .55+.02i  .02+.8i  |
      .29+.18i  .29+.2i   .27+.88i |
      .99+.43i  .38+.45i  .24+.08i |
      .17+.36i  .05+i     .084+.2i |
      .12+.094i .4+.51i   .91+.87i |
      .89+.98i  .17+.057i .75+.99i |
      .94+.03i  .79+.64i  .63+.71i |
      .26+.089i .87+.23i  .54+.68i |
      .19+.26i  .12+.71i  .45+.61i |
      .81+.39i  .83+.68i  .59+.71i |

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

o14 = | .25+.49i .55+.16i |
      | .5+.29i  .35+.27i |
      | .52+.8i  .86+.73i |
      | .32+.21i .68+.84i |
      | .49+.78i .2+.06i  |
      | .12+.6i  .76+.63i |
      | .28+.6i  .76+.77i |
      | .76+.05i .93+.57i |
      | .47+i    .16+.75i |
      | .06+.57i .69+.36i |

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

o15 = | 2.4+.77i  1.6-i     |
      | -2-1.9i   -1.6+1.2i |
      | 2.6+.74i  1.4-1.8i  |
      | -.75+2.7i 1.2+.59i  |
      | -.45+2.4i 1.3+.79i  |
      | -1.2-1.7i -1+.82i   |
      | -1.3-.35i -.15+.96i |
      | 1.6-.69i  -.05-1.5i |
      | -.5+.62i  .59+.73i  |
      | .36-1.3i  -1.5-.35i |

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

o16 = 2.00148302124336e-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 = | .51 .65 .22 .12  .067 |
      | .57 .15 .97 .22  .75  |
      | .41 .23 .77 .65  .65  |
      | .81 .9  .37 .023 .081 |
      | .43 .39 .43 .21  .019 |

                 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 = | -40 -8.8 11   29   -5.1 |
      | 27  5.7  -7.2 -19  2.6  |
      | 20  5.6  -6.8 -16  5.6  |
      | -11 -3.9 4.8  7.3  -.73 |
      | 2.1 .78  .55  -.18 -3.6 |

                 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 = 7.105427357601e-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 = | -40 -8.8 11   29   -5.1 |
      | 27  5.7  -7.2 -19  2.6  |
      | 20  5.6  -6.8 -16  5.6  |
      | -11 -3.9 4.8  7.3  -.73 |
      | 2.1 .78  .55  -.18 -3.6 |

                 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 :