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Macaulay2Doc :: solve

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 = | .066+.026i .03+.62i .08+.8i  .12+.5i  .65+.33i .6+.83i   .13+.19i
      | .61+.99i   .23+.81i .4+.81i  .53+.7i  .82+.51i .62+.42i  .97+.22i
      | .26+.53i   .54+.91i .75+.87i .77+.52i .31+.31i 1+.28i    .71+.72i
      | .42+.26i   .99+.57i .85+.26i .56+.49i .67+.17i .63+.14i  .08+.77i
      | .19+.62i   .95+.7i  .8+.64i  .3+.06i  .04+.6i  .19+.22i  .001+.3i
      | .22+.85i   .88+.83i .53+.05i .92+.05i .98+.75i .26+.083i .33+.99i
      | .99+.39i   .17+.17i .55+.71i .36+.34i .92+.57i .32+.37i  .05+.75i
      | .007+.36i  .05+.6i  .98+.96i .84+.99i .24+.41i .82+.23i  .54+.27i
      | .025+.22i  .46+.7i  .99+.17i .28+.3i  .41+.94i .54+.1i   .42+.65i
      | .41+.42i   .24+.16i .64+.73i .21+.47i .63+.18i .073+.46i .73+.76i
      -----------------------------------------------------------------------
      .11+.26i  .85+.28i  .18+.27i |
      .84+.65i  .53+.04i  .67+.04i |
      .9+.46i   .086+.31i .22+.71i |
      .18+i     .04+.81i  .3+.75i  |
      .67+.62i  .02+.66i  .21+.88i |
      .1+.57i   .058+.2i  .38+.54i |
      .06+.94i  .95+.89i  .73+.31i |
      .16+.018i .5+.9i    .32+.94i |
      .43+.95i  .13+.25i  .25+.78i |
      1+.59i    .48+.29i  .61+.76i |

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

o14 = | .068+.3i .02+.81i |
      | .03+.96i .76+.08i |
      | .52+.91i .27+.77i |
      | .69+.13i .98+.77i |
      | .67+.91i .47+.34i |
      | .13+.16i .28+.84i |
      | .23+.95i .73+.28i |
      | .85+.83i .63+.58i |
      | .1+.1i   .62+.78i |
      | .83+.36i .57+.19i |

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

o15 = | 2.2-1.1i  -.87+.12i  |
      | .1-1.6i   .47+.53i   |
      | .39+1.7i  .46-.66i   |
      | -1.5-.86i -.079-.25i |
      | -.77+1.4i .37+.067i  |
      | -.46+.64i -.048+.44i |
      | .23-.54i  .32+.34i   |
      | -.14+1.2i -.2-.72i   |
      | 2+.25i    -.35-.38i  |
      | -1.6-2i   .74+.71i   |

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

o16 = 1.96732021192296e-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 = | .64  .7  .84  .85 .073 |
      | .21  .18 .79  .68 .95  |
      | .57  .45 .77  .45 .83  |
      | .57  .53 .053 .76 1    |
      | .017 .82 .8   .27 .45  |

                 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 = | .29   -1.2 2    -.067 -1   |
      | -.006 -.97 -.29 .62   1.2  |
      | .17   .46  .65  -1    .045 |
      | .86   1.2  -1.9 .51   -.28 |
      | -.83  .24  .45  .38   .17  |

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

o20 = 2.22044604925031e-16

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

o21 = 7.49400541621981e-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 = | .29   -1.2 2    -.067 -1   |
      | -.006 -.97 -.29 .62   1.2  |
      | .17   .46  .65  -1    .045 |
      | .86   1.2  -1.9 .51   -.28 |
      | -.83  .24  .45  .38   .17  |

                 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 :