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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 = | .98+.89i .01+.9i   .89+.55i  .19+.94i .16+.6i   .05+.26i .03+.38i
      | .31+.38i .44+.22i  .88+.97i  .22+.41i .07+.82i  .08+.63i .37+.44i
      | .94+.44i .46+.65i  .17+.7i   .63+.93i .34+.41i  .83+.2i  .5+.79i 
      | .08+.82i .86+.1i   .82+.12i  .18+.46i .07+.65i  .53+.85i .12+.78i
      | .68+.68i .48+.74i  .84+.45i  .25+.72i .52+.5i   .51+.2i  .94+.45i
      | .88+.71i .63+.85i  .73+.71i  .25+.71i .39+.046i .23+.58i .86+.18i
      | .47+.96i .32+.083i .39+.94i  .5+.89i  .35+.76i  .28+.62i .92+.58i
      | .63+i    .69+.48i  .042+.13i .93+.16i .35+.24i  .85+.26i .88+.46i
      | .85+.71i .01+.68i  .15+.84i  .17+.87i .99+.04i  .9+.92i  .79+.77i
      | .94+.42i .4+.059i  .68+.53i  .91+.47i .74+.44i  .69+.91i .78+.07i
      -----------------------------------------------------------------------
      .54+.79i .78+.79i  .28+.94i |
      .83+.83i .08+.92i  .37+.88i |
      .75+.01i .98+.88i  .24+.29i |
      .85+.49i .25+.18i  .4+i     |
      .4+.33i  .52+.55i  .25+.41i |
      .25+.91i .9+.77i   .46+.21i |
      .26+.33i .05+.62i  .85+.36i |
      1+.69i   .14+.006i .12+.43i |
      .24+.18i .73+.66i  .33+.1i  |
      .91+.89i .9+.92i   .85+.9i  |

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

o14 = | .87+.26i  .22+.6i  |
      | .12+.62i  .34+.63i |
      | .85+.09i  .18+.33i |
      | .13+.19i  .88+.28i |
      | .55+.71i  .87+.63i |
      | .73+.35i  .94+.19i |
      | .049+.27i .29+.14i |
      | .48       .3+.81i  |
      | .88+.33i  .52+.31i |
      | .23+.42i  .78+.26i |

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

o15 = | -.28-.41i .053-.025i |
      | .38+.44i  .93-.6i    |
      | .11-.33i  -.11+.19i  |
      | -.18-.97i -.7+.48i   |
      | 1.6+.99i  .19-.44i   |
      | -.62+1.1i .55-.33i   |
      | -.65-.35i .15+.69i   |
      | .9-.43i   -.56+.28i  |
      | .15+.006i .098-.12i  |
      | -.29+.33i .43-.5i    |

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

o16 = 7.06677886012864e-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 = | .59 .85 .42 .39 .38 |
      | .85 .54 .51 .2  .43 |
      | .29 .17 .7  .41 .76 |
      | .25 .66 .7  .72 .19 |
      | .69 .49 .55 .51 .5  |

                 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 = | -1.2 .86   -1.1 -.41  2    |
      | 2.3  -.096 .16  .088  -1.9 |
      | -1.8 3.1   .85  2     -3.4 |
      | -.4  -2.8  -.97 -.094 4.3  |
      | 1.8  -1.7  1.4  -1.7  .54  |

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

o20 = 4.9960036108132e-16

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

o21 = 4.71844785465692e-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 = | -1.2 .86   -1.1 -.41  2    |
      | 2.3  -.096 .16  .088  -1.9 |
      | -1.8 3.1   .85  2     -3.4 |
      | -.4  -2.8  -.97 -.094 4.3  |
      | 1.8  -1.7  1.4  -1.7  .54  |

                 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 :