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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 = | .83+.3i   .83+.29i   .21+.6i  .82+.5i    .93+.8i  .19+.016i .9+.68i 
      | 1+.08i    .091+.034i .32+.97i .47+.23i   .22+.42i .95+.23i  .85+.17i
      | .6+.4i    .69+.38i   .88+.51i .57+.35i   .74+.49i .18+.58i  .48+.27i
      | .85+.94i  .26+.98i   .56+.98i .04+.96i   .36+.39i .95+.55i  .32+.27i
      | .1+.89i   .97+.42i   .74+.97i 1+.07i     .19+.22i .68+.45i  .54+.96i
      | .26+.11i  .065+.25i  .87+.44i .033+.089i .18+i    .51+.45i  .27+.38i
      | .087+.13i .29+.41i   .65+.34i .33+.21i   .58+.49i .37+.84i  .51+.41i
      | .83+.66i  .64+.47i   .33+.25i .92+.72i   .18+.48i .85+.01i  .19+.77i
      | .62+.38i  .61+.58i   .21+.58i .11+.012i  .5+.91i  .28+.19i  .71+.79i
      | .18+.85i  .26+.085i  .58      .2+.71i    .72+.79i .93+.19i  .78+.55i
      -----------------------------------------------------------------------
      .092+.12i 1+.54i   .097+.016i |
      .87+.09i  .54+.86i .79+.45i   |
      .46+.79i  .68+.81i .62+.66i   |
      .54+.62i  .13+.83i .28+.9i    |
      .95+.68i  .43+.41i .35+.5i    |
      .88+.14i  .6+.61i  .08+.67i   |
      .98+.77i  .16+.54i .59+.29i   |
      .78+.49i  .01+.85i .48+.28i   |
      .96+.96i  .28+.54i .02+.73i   |
      .2+.47i   .04+.18i .07+.91i   |

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

o14 = | .49+.42i .51+.2i  |
      | .31+.33i .37+.4i  |
      | .46+.27i .98+.7i  |
      | .95+.6i  .6+.05i  |
      | .27+.36i .52+.72i |
      | .57+.29i .11+.41i |
      | .21+.78i .89+i    |
      | .98+.86i .43+.58i |
      | .15+.77i .52+.91i |
      | .8+.04i  .38+.18i |

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

o15 = | .51+.5i     -.04-.1i  |
      | -.98+1.5i   1.3i      |
      | 1.8-.71i    1.1-.74i  |
      | -1.2-.09i   -1.8+.59i |
      | 2.6+2.1i    2.3+1.3i  |
      | .58+1.1i    .6+.71i   |
      | -1.3-2.6i   -.91-1.6i |
      | -.064+.067i -.28-.53i |
      | -.71-1.9i   -.45-1.7i |
      | -1.3+.55i   .02+1.2i  |

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

o16 = 2.00148302124336e-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 = | .63  .18 .079 .27 .63  |
      | .026 .32 .22  .11 .099 |
      | .93  .34 .92  .34 .84  |
      | .72  .97 .24  .79 .67  |
      | .42  .77 .49  .33 .24  |

                 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 = | .8   -5.9 -.21 -.78 3.3  |
      | 2.3  2.5  -1.5 -1.3 2    |
      | -2.1 .25  1.5  .31  -.65 |
      | -5   -3.4 1.9  3.9  -3.2 |
      | 2.5  6.6  -.33 -.54 -2.5 |

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

o20 = 1.11022302462516e-15

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

o21 = 8.88178419700125e-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 = | .8   -5.9 -.21 -.78 3.3  |
      | 2.3  2.5  -1.5 -1.3 2    |
      | -2.1 .25  1.5  .31  -.65 |
      | -5   -3.4 1.9  3.9  -3.2 |
      | 2.5  6.6  -.33 -.54 -2.5 |

                 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 :