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 = | -2.2e-16 |
      | -2.2e-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 = | .3+.65i   .54+.65i  .28+.37i .78+.72i  .36+.79i .73+.51i .49+.99i
      | .02+.85i  .33+.76i  .4+.71i  .44+.13i  .82+.84i .26+.12i .5+.008i
      | .85+.55i  .93+.7i   .95+.58i .56+.27i  .54+.14i .9+.35i  .66+.4i 
      | .64+.34i  .63+.89i  .95+.86i .032+.24i .78+.92i .36+.57i .95+.39i
      | .53+.49i  .84+.97i  .48+.19i .9+.27i   .99+.55i .22+.72i .62+.14i
      | .84+.97i  .063+.18i .51+.57i .81+.89i  .18+.71i .3+.11i  .47+.16i
      | .39+.59i  .96+.91i  .13+.79i .76+.86i  .95+.34i .71+.28i .6+.61i 
      | .076+.36i .13+.11i  .42+.66i .2+.86i   .5+.17i  .91+.6i  .79+.26i
      | .79+.49i  .81+.37i  .63+.73i .66+.06i  .85+.18i .18+.35i .39+.44i
      | .21+.72i  .65+.89i  .41+.2i  .52+.63i  .99+.61i .63+.38i .24+.71i
      -----------------------------------------------------------------------
      .08+.68i .73+.2i   .84+.48i |
      .17+.24i .54+.74i  .35+.73i |
      .53+.45i .55+.42i  .12+.46i |
      .5+.92i  .61+.45i  .96+.91i |
      .44+.34i .19+.96i  .67+.51i |
      .78+.56i .2+.49i   .66+.41i |
      .7+.09i  .53+.27i  .59+.16i |
      .94+.15i .85+.74i  .73+.05i |
      .97+.6i  .041+.45i .31+.83i |
      .81+.95i .43+.081i .65+.34i |

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

o14 = | .28+.27i  .37+.88i |
      | .31+.75i  .32+.27i |
      | .85+.01i  1+.58i   |
      | .81+.12i  .15+.73i |
      | .007+.44i .51+.05i |
      | .8+.81i   .91+.35i |
      | .84+.32i  .33+.67i |
      | .98+.05i  .57+.15i |
      | .6+.08i   .68+.47i |
      | .53+.3i   .81+.3i  |

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

o15 = | .48+.38i  .52+.79i  |
      | .2-.088i  -1.4      |
      | .52-.54i  .74-.21i  |
      | -.38-.34i .18-.5i   |
      | .1i       1.6+1.2i  |
      | -1+1.3i   2.4+2.3i  |
      | -.25-.9i  -.74-.55i |
      | .31-.57i  -1.4-i    |
      | 1.1-1.2i  -2.1-1.8i |
      | -.27+1.2i 1-.11i    |

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

o16 = 7.7715611723761e-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 = | .81 .33 .26  .95 .069 |
      | .25 .56 .77  .14 .2   |
      | .11 .84 .69  .35 .04  |
      | .6  .12 .016 .59 .31  |
      | .33 .54 .4   .54 .025 |

                 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 = | 64  -21 100 .67  -170 |
      | 44  -18 75  1.9  -120 |
      | -34 15  -58 -2.2 96   |
      | -57 19  -91 -.84 160  |
      | -32 11  -47 2.9  81   |

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

o20 = 2.8421709430404e-14

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

o21 = 8.88178419700125e-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 = | 64  -21 100 .67  -170 |
      | 44  -18 75  1.9  -120 |
      | -34 15  -58 -2.2 96   |
      | -57 19  -91 -.84 160  |
      | -32 11  -47 2.9  81   |

                 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 :