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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 = | .04+.74i .62+.88i  .14+.5i  .29+.54i  .9+.55i  .75+.47i .12+.33i
      | .45+.78i .57+.08i  .86+.77i .42+.025i .24+.83i .24+.26i .59+.22i
      | .12+.09i .52+.6i   .62+.5i  .023+.26i .9+.12i  .83+.34i .14+.59i
      | .51+.39i .03+.98i  .85+.71i .013+.43i .62+.18i .63+.48i .33+.2i 
      | .77+.55i .07+.93i  .07+.57i .7+.74i   .08+.65i .28+.41i .03+.71i
      | .13+.38i .04+.59i  .47+.7i  .31+.12i  .38+.18i .6+.69i  .58+.68i
      | .02+i    .26+.62i  .24+.54i .05+.84i  .6+.2i   .44+.77i .12+.65i
      | .05+.71i .067+.14i .97+.24i .86+.36i  .4+.53i  .05+.16i .57+i   
      | .87+.75i .11+.44i  .73+.44i .85+.5i   .86+.95i .36+.43i .91+.28i
      | .61+.94i .21+.73i  .08+.49i .08+.51i  .89+.16i .04+.85i .4+.32i 
      -----------------------------------------------------------------------
      .68+.64i .82+.26i .58+.77i  |
      .8+.93i  .82+.64i .73+.03i  |
      .51+.8i  .03+.92i .4+.05i   |
      .98+.54i .19+.52i .66+.36i  |
      .02+.72i .42+.13i .5+.14i   |
      .35+.47i .91+.3i  .1+.71i   |
      .13+.9i  .54+.77i .78+.86i  |
      .02+.51i .97+.44i .61+.76i  |
      .15+.28i .39+.17i .14+.025i |
      .27+.99i .73+.55i .44+.23i  |

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

o14 = | .1+.59i   .89+.34i  |
      | .88+.05i  .26+.93i  |
      | .17+.14i  .97+.88i  |
      | .29+.25i  .066+.33i |
      | .43+.009i .04+.77i  |
      | .46+.79i  .68+.06i  |
      | .11+.34i  .46+.39i  |
      | .6+.16i   .67+.02i  |
      | .69+.2i   .03+.76i  |
      | .51+.59i  .54+.51i  |

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

o15 = | -.3-2i    -.06+.71i  |
      | .69-.68i  .063+.32i  |
      | 2-.01i    -.14+.07i  |
      | .38+.63i  -.37-.21i  |
      | -.53+.81i .78-.09i   |
      | .18-.19i  .18+.36i   |
      | -.94+.21i -.11-.47i  |
      | -.49-.85i .011-.45i  |
      | -.19+1.1i .78+.05i   |
      | -1.4+.61i -.16-.043i |

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

o16 = 5.55111512312578e-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 = | .01  .63 .83 .81  .018 |
      | .97  .62 .51 .26  .3   |
      | .072 .62 .56 .33  .6   |
      | .08  .36 .31 .075 .97  |
      | .8   .67 .27 .55  .58  |

                 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 = | .34  1    -2   .83  .078 |
      | -2.7 -.81 6.8  -4.4 .84  |
      | 1.6  1.7  -2.2 2    -2.1 |
      | 1.7  -1.1 -2.9 1.3  1.4  |
      | .31  -.25 -1.4 1.8  .24  |

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

o20 = 1.25940924355916e-15

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

o21 = 1.11022302462516e-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 = | .34  1    -2   .83  .078 |
      | -2.7 -.81 6.8  -4.4 .84  |
      | 1.6  1.7  -2.2 2    -2.1 |
      | 1.7  -1.1 -2.9 1.3  1.4  |
      | .31  -.25 -1.4 1.8  .24  |

                 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 :