next | previous | forward | backward | up | top | index | toc | Macaulay2 web site
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 = | .5+.75i    .6+.66i   .15+.93i .89+.48i .017+.29i .76+.89i  .59+.06i
      | .096+.024i .19+.26i  .96+.1i  .57      .54+.44i  .3+.31i   .68+.45i
      | .019+.41i  .97+.93i  .56+.03i .73+.38i .49+.74i  .24+.71i  .27+.91i
      | .23+.47i   .16+.38i  .78+.23i .61+.45i .53+.58i  .09+.91i  .71+.68i
      | .41+.46i   .82+.86i  .45+.87i .17+.81i .34+.47i  .5+.47i   .26+.12i
      | .82+.41i   .46+.29i  .78+.92i .81+.47i .093+.18i .95+.15i  .72+.71i
      | .84+.59i   .96+.95i  .34+.63i .91+.51i .12+.099i .092+.29i .52+.84i
      | .49+.28i   .029+.11i .93+.29i .58+.86i .58+.77i  .91+.01i  .04+.94i
      | .96+.3i    .97+.1i   .49+.5i  .09+.63i .45+.79i  .89+.22i  .36+.17i
      | .08+.85i   .48+.66i  .56+.83i .1+.43i  .78+.19i  .036+.41i .82+.75i
      -----------------------------------------------------------------------
      .18+.85i  .027+.04i .53+.56i |
      .75+.75i  .95+.05i  .49+.89i |
      .43+.19i  .46+.53i  .14+.51i |
      .9+.36i   .39+.19i  .87+.12i |
      .11+.69i  .78+.1i   .63+.46i |
      .42+.32i  .25+.49i  .01+.87i |
      .65+.07i  .67+.08i  .51+.63i |
      .22+.052i .96+.71i  .19+.49i |
      .52+.84i  .53i      .55+.56i |
      .63+.56i  .47+.68i  .32+.98i |

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

o14 = | .7+.4i     .28+.57i |
      | .99+.58i   .01+.51i |
      | .8         .02+.75i |
      | .81+.62i   .25+.9i  |
      | .48+.42i   .05+.74i |
      | .26+.49i   .33+.92i |
      | .42+.22i   .1+.23i  |
      | .97+.86i   .89+.97i |
      | .039+.093i .29+.16i |
      | .25+.8i    .73+.98i |

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

o15 = | .82+.12i  .67-1.9i  |
      | -.69-.52i -.41+.37i |
      | .83+.21i  .8-.02i   |
      | .52+.61i  .75       |
      | 1.5+.36i  1.3-i     |
      | .01-i     -.84+.59i |
      | -1.4+.21i -.05+1.4i |
      | -.74+.42i -.53+.79i |
      | -.35+.45i -.26+.4i  |
      | .52-.98i  -.6-.4i   |

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

o16 = 1.30429058107358e-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 = | .5   .18 .11  .6   .88 |
      | .77  .26 .24  .046 .3  |
      | .37  .61 .062 .54  .11 |
      | .9   .24 .79  .66  .53 |
      | .075 .82 .63  .23  .75 |

                 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 = | -.036 1.3  .14  .0052 -.5 |
      | -.43  .51  .98  -.66  .62 |
      | -.87  -.57 -.66 1.2   .5  |
      | .39   -1.4 .9   .7    -.5 |
      | 1.1   .23  -.8  -.5   .44 |

                 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 = 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 = | -.036 1.3  .14  .0052 -.5 |
      | -.43  .51  .98  -.66  .62 |
      | -.87  -.57 -.66 1.2   .5  |
      | .39   -1.4 .9   .7    -.5 |
      | 1.1   .23  -.8  -.5   .44 |

                 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 :