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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 = | 0        |
      | -3.3e-16 |
      | -8.9e-16 |

                 3          1
o10 : Matrix RR    <--- RR
               53         53
i11 : norm oo

o11 = 8.88178419700125e-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 = | .17+.12i .3+.82i  .72+.09i .082+.34i .26+.93i  .21+.93i .74+.44i
      | .35+.46i .56+.8i  .34+.66i .85+.22i  .063+.33i .75+.27i .59+.27i
      | .17+.93i .15+.91i .66+.92i .84+.74i  .35+.71i  .68+.79i .08+.8i 
      | .24+.98i .32+.12i .93+.23i .25+.64i  .92+.14i  .21+.53i .04+.66i
      | .63+.53i .06+i    .35+.33i .51+.56i  .85+.72i  .75+.8i  .58+.82i
      | .51+.74i .9+.83i  .22+.87i .29+.97i  .81+.71i  .14+.82i .6+.32i 
      | .73+.7i  .97i     .15+.41i .29+.12i  .98+.01i  .46+.62i .41+.47i
      | .93+.88i .46+.2i  .65+.19i .53+.18i  .69+.99i  .41+.69i .23+.41i
      | .82+.79i .29+.75i .23+.37i .59+.29i  .23+.38i  .45+.6i  .34+.18i
      | .62+.28i .04+.68i .86+.22i .36+.99i  .18+.18i  .21+.53i .5+.22i 
      -----------------------------------------------------------------------
      .24+.084i .51+.72i .51+.25i |
      .37+.87i  .4+.17i  .66+.33i |
      .44+.54i  .95+.35i .99i     |
      .62+.26i  .34+.24i .59+.09i |
      .71+.57i  .3+.94i  .84+.53i |
      .47+.079i .5+.02i  .044+.1i |
      .35+.83i  .78+.31i .96+.22i |
      .71+.66i  .53+.6i  .96+.53i |
      .82+.59i  .36+.27i .7+.59i  |
      .17+.21i  .33+.73i .74i     |

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

o14 = | .1+.63i  .8+.62i  |
      | .88+.66i .43+.13i |
      | .04+.52i .47+.57i |
      | .78+.16i .29+.67i |
      | .92+.39i .56+.13i |
      | .97+.26i .12+.99i |
      | .17+.84i .76+.39i |
      | .39+.64i .43+.7i  |
      | .51+.76i .64+.63i |
      | .47+.8i  .32+.48i |

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

o15 = | .041+.33i .16-.18i   |
      | -.23-.79i -.82-.55i  |
      | .35+.26i  .72-.06i   |
      | .93-1.2i  .28+.36i   |
      | .38-.68i  -.028-.14i |
      | -.04+.64i .75-1.1i   |
      | -.52+.43i -.28+.88i  |
      | .29-.32i  -.29+.34i  |
      | -.86+.46i -.32+.43i  |
      | .4+i      .25+.59i   |

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

o16 = 8.67111901826274e-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 = | .77  .15 .69 .74  .13 |
      | .7   .7  .5  .082 .75 |
      | .086 .1  .89 .11  .91 |
      | .88  .71 .37 .4   .85 |
      | .9   .31 .12 .73  .18 |

                 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 = | -6.5 9.7  3.8  -14  13   |
      | 7.5  -8.4 -5.4 15   -14  |
      | 2    .094 -.4  .53  -2.3 |
      | 5.2  -8.7 -2.8 12   -8.9 |
      | -2.8 .99  2.1  -2.3 3.7  |

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

o20 = 1.77635683940025e-15

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

o21 = 8.88178419700125e-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 = | -6.5 9.7  3.8  -14  13   |
      | 7.5  -8.4 -5.4 15   -14  |
      | 2    .094 -.4  .53  -2.3 |
      | 5.2  -8.7 -2.8 12   -8.9 |
      | -2.8 .99  2.1  -2.3 3.7  |

                 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 :