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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 = | .01+.74i .72+.24i  .64+.03i .03+.25i  .22+.61i   .87i     .17+.99i  
      | .88+.08i .2+.27i   .89+.69i .35+.38i  .95+.57i   .92+.6i  .14+.67i  
      | .47+.39i .21+.057i .56+.85i .94+.64i  .023+.093i .42+.13i .74+.79i  
      | .26+.66i .75+.66i  .12+.54i .036+.25i .81+.98i   .63+.37i .96+.07i  
      | .44+.26i .99+.47i  .38+.23i .84+.98i  .63+.82i   .64+.56i .3+.74i   
      | .38+.64i .74+.02i  .36+.6i  .78+.77i  .93+.88i   .89+.88i .72+.68i  
      | .31+.71i .027+.22i .71+.06i .26+.18i  .98+.15i   .54+.35i .13+.71i  
      | .52+.37i .94+.24i  .55+.68i .65+.28i  .98+.71i   .68+.86i .093+.061i
      | .86+.82i .41+.61i  .21+.27i .33+.84i  .25+.63i   .66+.22i .89+.04i  
      | .57      .93+.99i  .21+.9i  .57+.75i  .55+.67i   .81+.11i .92+.46i  
      -----------------------------------------------------------------------
      .68+.86i .34+.16i .07+.94i |
      .81+.29i .24+.36i .18+.46i |
      .26+.37i .4+.7i   .44+.72i |
      .75+.94i .32+.91i .99+.26i |
      .63+.98i .51+.96i .86+.76i |
      .27+.9i  .75+.67i .13+.73i |
      .75+.79i .45+.71i .65+.54i |
      .99+.36i .85+.23i .1+.88i  |
      .52+.81i .17+.43i .14+.64i |
      .26+.27i .52+.03i .78+.91i |

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

o14 = | .81+.56i  .73+.03i |
      | .59+.61i  .55+.14i |
      | .94+.42i  .83+.2i  |
      | .006+.28i .75+.72i |
      | .24+.45i  .83+.55i |
      | .51+.33i  .54+.9i  |
      | .68+.47i  .54+.03i |
      | .52+.96i  .39+.21i |
      | .47+.73i  .71+.35i |
      | .36+.23i  .38+.62i |

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

o15 = | 1.2-.77i  -.19+.34i   |
      | .62+.26i  .11+.75i    |
      | .18+.24i  .36-.38i    |
      | .07+.88i  -.58-.37i   |
      | .41-.18i  -.65-.23i   |
      | -.7+.67i  .42+.89i    |
      | .1-.69i   .016-.094i  |
      | -.56+.77i .76-.85i    |
      | -.04-.55i .39-.007i   |
      | -.29-.65i -.049-.074i |

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

o16 = 7.71185559270127e-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 = | .76  .19 .82 .89 .6   |
      | .9   .6  .24 .52 .85  |
      | .99  .1  .13 .51 .059 |
      | .058 .99 .94 .46 .36  |
      | .62  .75 .97 .53 .89  |

                 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   -.63 1.5  -.68 1.5  |
      | -.8  .79  .27  1.2  -.72 |
      | -.29 -1.8 .61  -.5  2.1  |
      | 2.1  1.4  -.97 1.3  -3.3 |
      | .4   .95  -1.3 -.77 .35  |

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

o20 = 4.44089209850063e-16

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

o21 = 6.10622663543836e-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   -.63 1.5  -.68 1.5  |
      | -.8  .79  .27  1.2  -.72 |
      | -.29 -1.8 .61  -.5  2.1  |
      | 2.1  1.4  -.97 1.3  -3.3 |
      | .4   .95  -1.3 -.77 .35  |

                 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 :