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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 = | .15+.21i  .75+.62i .89+.76i  .32+.78i  .33+.87i .41+.24i .46+.81i
      | .2+.26i   .81+.17i .45+.21i  .51+.56i  .42+.13i .39+.68i .91+.34i
      | .92+.11i  .42+.58i .41+.36i  .05+.59i  .28+.13i .52+.99i .22+.3i 
      | .42+.23i  .61+.23i .098+.43i .67+.81i  .18+.26i .12+.58i .64+.39i
      | .96+.49i  .93+.61i .86+.61i  .72+.1i   .69+.19i .001+.4i .87+.31i
      | .9+.06i   .8+.77i  .65+.67i  .49+.6i   .54+.31i .13+.5i  .02+.89i
      | .57+.2i   .13+.47i .26+.17i  .16+.04i  .2+.19i  .74+.73i .57+.75i
      | .51+.24i  .92+.86i .65+.88i  .069+.17i .97+.51i .21+.97i .32+.49i
      | .06+.77i  .78+.12i .79+.54i  .54+.99i  .88+.85i .38+.25i .63+.32i
      | .068+.14i .38+.13i .64+.91i  .81+.65i  .48+.51i .68+.21i .08+.6i 
      -----------------------------------------------------------------------
      .56+.54i .37+.83i .51+.47i  |
      .63+.79i .85+.9i  .53+.89i  |
      .61+.01i .1+.81i  .53+.47i  |
      .22+.49i .8+.25i  .38+.53i  |
      .63+.86i .65+.7i  .24+.71i  |
      .52+.06i .76+.29i .91+.33i  |
      .45+.43i .87+.89i .073+.14i |
      .47+.97i .19+.65i .18+.41i  |
      .25+.86i .61+.45i .73+.6i   |
      .68+.33i .48+.95i .51+.98i  |

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

o14 = | .34+.19i  .06+.51i |
      | .54+.35i  .59+.53i |
      | .73+.29i  .16+.48i |
      | .9+.3i    1+.5i    |
      | .79+.62i  .34+.4i  |
      | .39+.77i  .85+.32i |
      | .088+.36i .78+.96i |
      | .63+.09i  .5+.88i  |
      | .05+.76i  .14+.53i |
      | .99+.46i  .61+.24i |

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

o15 = | .53+2.7i  -.73+.11i |
      | -2.7+1.7i -.36-.92i |
      | -.04-4.1i .69+.2i   |
      | -1.2-1.3i .79-.55i  |
      | 1.7+1.3i  .06+.61i  |
      | -2.3+.16i .36-.96i  |
      | 2.7+.91i  -.6+.09i  |
      | .01+.22i  .36+.085i |
      | .65-1.6i  1.1+1.1i  |
      | .94+1.4i  -1.1-.23i |

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

o16 = 1.52731344151809e-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 = | .34 .72   .65 .23  .91 |
      | .63 .16   .82 .015 .37 |
      | .83 .14   .95 .17  .91 |
      | .14 .84   .44 .5   .33 |
      | .82 .0059 .33 .7   .72 |

                 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 = | 5.8  8.3  -10 -4.8 3.7  |
      | 4.5  5.1  -7  -2.4 1.8  |
      | -5.4 -5.1 8.3 4.2  -3.1 |
      | -4.8 -5.1 6.4 4    -1.2 |
      | .61  -2.1 1.7 -.41 -.27 |

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

o20 = 3.5527136788005e-15

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

o21 = 6.21724893790088e-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 = | 5.8  8.3  -10 -4.8 3.7  |
      | 4.5  5.1  -7  -2.4 1.8  |
      | -5.4 -5.1 8.3 4.2  -3.1 |
      | -4.8 -5.1 6.4 4    -1.2 |
      | .61  -2.1 1.7 -.41 -.27 |

                 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 :