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 = | .97+.44i .2+.71i  .94+.27i  .05+.66i .008+.5i .37+.13i  .42+.38i 
      | .42+.77i .08+.55i .3+.91i   .5+.11i  .72+.84i .47+.14i  .83+.16i 
      | .26+.21i .23+.55i .12+.49i  .72+.07i .13+.86i .31+.21i  .031+.46i
      | .53+.84i .6+.9i   .28+.54i  .81+.42i .98+.48i .36+.82i  .35+.57i 
      | .51+.47i .33+.67i .1+.96i   .59+.91i .2+.6i   .9+.14i   .23+.82i 
      | .83+.28i .87+.53i .71+.78i  .11+.96i .25+.41i .53+.02i  .4+.52i  
      | .58+.99i .8+.68i  .52+.75i  .78+.95i .56+.09i .25+.52i  .76+.52i 
      | .09+.87i .18+.69i .24+.078i .85+.16i .47+.51i .73+.78i  .3+.51i  
      | .72+.53i .08+.84i .67+.77i  .48+.8i  .3+.95i  .22+.061i .63+.58i 
      | .82+.67i 1+.78i   .14+.43i  .33+.58i .94+.37i .91+.17i  .54+.97i 
      -----------------------------------------------------------------------
      .32+.59i  .99+.56i .62+.7i    |
      .72+.22i  .09+.68i .98+.53i   |
      .81+.54i  .3+.99i  .81+.15i   |
      .31+.056i .45+.47i .3+.85i    |
      .81+.65i  .84+.76i .056+.078i |
      .51+.36i  .21+.5i  .04+.57i   |
      .86+.62i  .49+.11i .27+.2i    |
      .88+.29i  .43+.48i .63+.79i   |
      .78+.65i  .26+.8i  .63+.74i   |
      1+.68i    .62+.99i .14+.41i   |

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

o14 = | .53+.43i .94+.71i  |
      | .15+.34i .3+.68i   |
      | .98+.86i .28+.34i  |
      | .2+.11i  .76+.23i  |
      | .5+.54i  .11+.78i  |
      | .27+.99i .018+.49i |
      | .83+.33i .44+.63i  |
      | .59+.58i .087+.43i |
      | .87+.63i .51+.44i  |
      | .43+.2i  .52+.65i  |

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

o15 = | .14-.49i  1.4-.63i   |
      | -.17+.36i -.19+.76i  |
      | .79+.48i  -.38+.072i |
      | .42-.24i  .22+.3i    |
      | -.45+.12i .76-.53i   |
      | -.58+.11i -.85+.33i  |
      | -.14-2i   -1+.41i    |
      | .2+.8i    .17-.65i   |
      | -.31-.22i .71+.48i   |
      | .8+.9i    -.15-.45i  |

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

o16 = 9.55049957678547e-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 = | .83  .25 .72  .95 .83 |
      | .047 .82 .59  .95 .99 |
      | .19  .13 .046 .1  .65 |
      | .72  .7  .56  .74 .84 |
      | .61  .68 .89  .56 .54 |

                 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 = | .05  -1.2 -.25 1.8  -.36  |
      | -1.6 .1   -.76 2.1  -.043 |
      | .54  .18  .68  -2.7 2.2   |
      | .91  .69  -1.8 .74  -1.6  |
      | .12  .21  2    -.86 .21   |

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

o20 = 4.44089209850063e-16

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

o21 = 5.55111512312578e-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 = | .05  -1.2 -.25 1.8  -.36  |
      | -1.6 .1   -.76 2.1  -.043 |
      | .54  .18  .68  -2.7 2.2   |
      | .91  .69  -1.8 .74  -1.6  |
      | .12  .21  2    -.86 .21   |

                 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 :