next | previous | forward | backward | up | top | index | toc | Macaulay2 web site

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 = | 0        |
      | -3.3e-16 |
      | -8.9e-16 |

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

o11 = 8.88178419700125e-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 = | .62+.93i  .71+.53i  .46+.052i .93+.61i  .36+.62i   .27+.18i 
      | .33+.36i  .29+.21i  .76+.47i  .92+.49i  .55+.76i   .61+.02i 
      | .081+.21i .04+.94i  .56+.34i  .87+.07i  .96+.64i   .15+.061i
      | .41+.36i  .78+.99i  .13+.71i  .4+i      .15+.61i   .94+.31i 
      | .15+.045i .57+.12i  .55+.93i  .97+.31i  .071+.094i .69+.56i 
      | .1+.2i    .23+.47i  .92+.74i  .36+.94i  .69+.42i   .64+.22i 
      | .25+.22i  .2+.12i   .85+.76i  .26+.5i   .77+.11i   .4+.23i  
      | .53+.67i  .69+.83i  .83+.5i   .26+.97i  .58+.51i   .92+.64i 
      | .29+.12i  .045+.41i .42+.23i  .096+.26i .085+.29i  .19+.95i 
      | .59+.22i  .066+.23i .56+.31i  .24+.4i   .42+.97i   .11+.056i
      -----------------------------------------------------------------------
      .45+.52i  .6+.39i  .07+.68i .8+.38i    |
      .94+.36i  .91+.51i .72+.51i .74+.68i   |
      .42+.8i   .12+.5i  .7+.51i  .14+.51i   |
      .03+.056i .11+.68i .64+.6i  .57+.86i   |
      .87+.91i  .67+.2i  .46+.51i .57+.3i    |
      .39+.19i  .79+.83i .74+.38i .006+.097i |
      .44+.49i  .99+.18i .87+.97i .29+.62i   |
      .16+.58i  .19+.99i .52+.26i .65+.35i   |
      .96+.6i   .07+.77i .12+.97i .59+.61i   |
      .92+.23i  .55+.97i .46+.14i .096+.35i  |

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

o14 = | .37+.98i .81+.6i  |
      | .91+.37i .39+.66i |
      | .79+.07i .07+.87i |
      | .67+.71i .45+.5i  |
      | .14+.49i .73+.76i |
      | .6+.91i  .21+.24i |
      | .94+.94i .31+.19i |
      | .35+.19i .68+.1i  |
      | .06+.84i .03+.37i |
      | .33+.71i .33+.29i |

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

o15 = | -.48-.027i .19-.36i   |
      | .56+i      .78-.31i   |
      | .71i       .21-.44i   |
      | .53-.42i   -.32+.36i  |
      | .6-.66i    -.034-.48i |
      | -.18-.75i  -.032-.28i |
      | -.22+.39i  .24+.036i  |
      | -.44+.34i  .22+.65i   |
      | 1.1-.24i   -.14+.26i  |
      | -.45-.5i   .022+.28i  |

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

o16 = 6.95552731308039e-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 = | .78 .69 .17 .04 .92 |
      | .17 .68 .33 .67 .28 |
      | .61 .62 1   .38 .54 |
      | .83 .38 .63 .12 .68 |
      | .39 .83 .43 .25 .72 |

                 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 = | 7.8  -.075 7    -7.4 -8.3 |
      | 10   -1.4  11   -13  -7.9 |
      | -1.9 -.81  .27  .99  1.6  |
      | -6.4 2.9   -7.2 8.6  4.3  |
      | -13  1.2   -14  16   12   |

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

o20 = 1.77635683940025e-15

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

o21 = 3.5527136788005e-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 = | 7.8  -.075 7    -7.4 -8.3 |
      | 10   -1.4  11   -13  -7.9 |
      | -1.9 -.81  .27  .99  1.6  |
      | -6.4 2.9   -7.2 8.6  4.3  |
      | -13  1.2   -14  16   12   |

                 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 :