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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 = | .42+.6i  .23+.49i .1+.75i     .48+.55i .46+.67i .59+.19i .11+.98i
      | .88+.21i .13+.38i .4+.89i     .87+.8i  .63+.87i .49+.16i .66+.61i
      | .08+.68i .7+.74i  .98+.79i    .62+.76i .64+.14i .62+.12i .38+.95i
      | .77+.85i .96+.26i .27+.59i    .16+.04i .26+.72i .62+.89i .58+.17i
      | .37+.99i .25+.06i .93+.52i    .21+.84i .95+.66i .18+.29i .72+.45i
      | .89+.59i .11+.17i .56+.31i    .72+.92i .63+.98i .81+.37i .91+.18i
      | .75+.04i .34+.15i .63+.63i    .66+.84i .86+.63i .96+.03i .9+.25i 
      | .97+.54i .09+.59i .69+.73i    .78+.57i 1+.83i   .57+.24i .23+.47i
      | .01+.83i .89+.23i .046+.0071i .08+.66i .97+.7i  .23+.15i .91+.2i 
      | .73+.03i .02+.57i .36+.79i    .39+.51i .55+.04i .03+.47i .46+.31i
      -----------------------------------------------------------------------
      .81+.17i .68+.07i .31+.83i |
      .88+.79i .9+.26i  .92+.18i |
      .71+.25i .99+.14i .73+.74i |
      .44+.63i .51+.18i .33+.84i |
      .78+.4i  .45+.99i .13+.38i |
      .84+.4i  .16+.9i  .84+.79i |
      .95+.18i 1+.36i   .42+.62i |
      .48+.79i .84+.76i .92+.63i |
      .78+.34i .99+.12i .6+.95i  |
      .86+.75i .17+.85i .91+.95i |

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

o14 = | .11+.1i  .94+.29i   |
      | .62+.82i .68+.63i   |
      | .21+.86i .53+.33i   |
      | .68+.72i .66+.03i   |
      | .97+.17i .097+.011i |
      | .36+.97i .17+.53i   |
      | .63+i    .84+.51i   |
      | .35+.51i .08+.94i   |
      | .56+.14i .3+.76i    |
      | .78+.47i .47+.74i   |

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

o15 = | -.09-2.2i -5.7+1.5i |
      | -1.8-.46i -1.9+8.1i |
      | .39-1.1i  -2-.37i   |
      | -1.1+2i   5.2+4.5i  |
      | -1.5+.55i 2.2+2.1i  |
      | .64+2.4i  2.5-4.6i  |
      | 1.1+1.7i  2-4.6i    |
      | 2-1.1i    -1.4-2.2i |
      | .87-.12i  1.4-1.9i  |
      | .063-.11i .69-1.5i  |

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

o16 = 4.45336459267844e-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 = | .24 .048 .98 .48  .38  |
      | .83 .74  .19 .43  .53  |
      | .52 .72  .38 .44  .79  |
      | .32 .025 .69 .41  .45  |
      | .88 .78  .64 .076 .021 |

                 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 = | -2.9 .61  -1.3 4    .7   |
      | 2.5  .091 1.2  -4.3 .08  |
      | .36  -1.4 .58  .3   .76  |
      | 5.3  4.3  -2.3 -5.5 -2.2 |
      | -3.5 -2.2 2.1  4.2  .28  |

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

o20 = 1.33226762955019e-15

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

o21 = 1.94289029309402e-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 = | -2.9 .61  -1.3 4    .7   |
      | 2.5  .091 1.2  -4.3 .08  |
      | .36  -1.4 .58  .3   .76  |
      | 5.3  4.3  -2.3 -5.5 -2.2 |
      | -3.5 -2.2 2.1  4.2  .28  |

                 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 :