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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 = | .39+.65i .5+.71i   .84+.95i  .37+.17i  .05+.93i .56+.13i  .9+.73i 
      | .26+.29i .73+.72i  .34+.46i  .26+.8i   .02+.87i .4+.3i    .98+.86i
      | .04+.91i .09+.61i  .55+.59i  .6+.38i   .27+.15i .26+.64i  .19+.62i
      | .01+.53i .53+.89i  .17+.69i  .25+.89i  .81+.37i .78+.68i  .1+.52i 
      | .97+.24i .44+.81i  .12+.97i  .91+.05i  .61+.66i .066+.16i .87+.23i
      | .59+.5i  .6+.64i   .24+.44i  .15+.44i  .84+.57i .54+.64i  .61+.96i
      | .8+.35i  .24+.11i  .16+.33i  .54+.12i  .41+.81i .21+.12i  .89+.72i
      | .58+.72i .45+.067i .074+.47i .06+.64i  .49+.39i .29+.9i   .54+.48i
      | .91+.41i .68+.57i  .41+.76i  .68+.97i  .92+.17i .49+.33i  .16+.3i 
      | .91+.19i .075+.24i .35+.71i  .25+.079i .35+.27i .49+.73i  .48+.95i
      -----------------------------------------------------------------------
      .56+.06i .38+.86i  .29+.99i |
      .03+.95i .72+.48i  .18+.68i |
      .88+.29i .023+.19i .44+.63i |
      .69+.21i .95+.96i  .76+.2i  |
      .54+.13i .69+.49i  .16+.92i |
      .79+.97i .62+.66i  .25+.13i |
      .85+.04i .82+.74i  .59+.17i |
      .36+.34i .71+.94i  .71+.64i |
      .22+.99i .9+.36i   .94+.24i |
      .42+.23i .78+.06i  .23+.89i |

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

o14 = | .72+.03i  .33+.39i  |
      | .72+.95i  .32+.47i  |
      | .54+.29i  .34+.31i  |
      | .35+.98i  .097+.23i |
      | .82+.35i  .37+.7i   |
      | .06+.97i  .1+.43i   |
      | .76+.26i  .5+.94i   |
      | .34+.37i  .45+.31i  |
      | .46+.44i  .77+.09i  |
      | .31+.035i .16+.22i  |

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

o15 = | .79-.5i   1.5-.02i  |
      | 1.4-.4i   1.4-.21i  |
      | -.46-i    .42-1.5i  |
      | -1.1-.8i  -1.5-1.8i |
      | -1.7-.92i -1.2-1.5i |
      | -.96-.04i -1.6-.87i |
      | -.23+.56i -.22+.59i |
      | 1.5+.68i  1.1+1.2i  |
      | .49+1.6i  -.47+1.8i |
      | .42+.53i  .3+1.4i   |

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

o16 = 1.93892115658267e-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 = | .73 .052 .98 .25 .11 |
      | .37 .72  .58 .75 .13 |
      | .27 .38  .4  .87 .98 |
      | .64 .53  .77 .89 .12 |
      | .61 .45  .49 .4  .86 |

                 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 = | -1.6 -3.1 -1.8 3.8  2.2  |
      | -.34 2.4  -.84 -1.5 .85  |
      | 2.4  2.5  1.1  -3.2 -1.5 |
      | -.77 -1.4 .73  2.1  -.82 |
      | .28  .15  .75  -1.1 .41  |

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

o20 = 6.66133814775094e-16

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

o21 = 5.27355936696949e-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 = | -1.6 -3.1 -1.8 3.8  2.2  |
      | -.34 2.4  -.84 -1.5 .85  |
      | 2.4  2.5  1.1  -3.2 -1.5 |
      | -.77 -1.4 .73  2.1  -.82 |
      | .28  .15  .75  -1.1 .41  |

                 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 :