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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 |
      | 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 = | .21+.9i  .91+.65i   .3+.64i  .96+.17i .043+.037i .12+.064i  .038+.15i
      | .05+.58i .26+.52i   .21+.98i .34+.41i .34+.56i   .092+.015i .82+.07i 
      | .59+.68i .57+.88i   .12+.03i .82+.89i .77+.23i   .73+.08i   .95+.04i 
      | .23+.58i .42+.19i   .39+.95i .96+.16i .29+.77i   .15+.8i    .067+.49i
      | .2+.87i  .51+.83i   .5+.43i  .97+.65i .47+.075i  .17+.18i   .83+.05i 
      | .06+.72i .71+.16i   .94+.41i .32+.64i .88+.93i   .45+.52i   .11+.18i 
      | .81+.03i .095+.022i .11+.97i .67+.68i .88+.74i   .25+.61i   .96+.29i 
      | .88+.88i .95+.68i   .97+.75i .67+.99i .32+.53i   .68+.58i   .47+.56i 
      | .45+.53i .63+.21i   .93+.71i .87+.88i .53+.95i   .89+.39i   .61+.79i 
      | .29+.81i .88+.25i   .75+.5i  .35+.23i .23+.48i   .46+.29i   .24+.058i
      -----------------------------------------------------------------------
      .61+.57i .12+.65i .61+.54i |
      .63+.32i .62+.58i .69+.65i |
      .05+.55i 1+.85i   .95+.16i |
      .55+.35i .33+i    .08+.97i |
      .56+.21i .72+.76i .53+.03i |
      .84+.57i .84+.18i .14+.64i |
      .62+.86i .65+.33i .77+.34i |
      .34+.67i .85+.34i .94+.11i |
      .07+.66i .56+.58i .42+.46i |
      .29+.38i .24+.79i .51+.72i |

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

o14 = | .22+.71i  .48+.11i  |
      | .059+.32i 1+.37i    |
      | .5+.32i   .95+.2i   |
      | .74+.27i  .8+.27i   |
      | 1+.87i    .08+.6i   |
      | .011+.49i .64+.47i  |
      | .46+.63i  .029+.14i |
      | .69+.81i  .84+.83i  |
      | .27+.21i  .14+.4i   |
      | .43+.38i  .11+.67i  |

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

o15 = | .01+1.1i  4.4+2.8i  |
      | .72+.48i  -.29-4.2i |
      | .59+.48i  .94-.03i  |
      | .17-.9i   -.77-.44i |
      | -.93+1.6i .98-1.2i  |
      | -.68-1.2i -1.1+3.5i |
      | .94-.12i  -2.1-.55i |
      | .23+.21i  .97+1.6i  |
      | .63-.52i  .43-.65i  |
      | -.29-.86i -1.7-2.1i |

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

o16 = 1.54636173612489e-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 = | .41 .72 .11 .83  .45 |
      | .57 .37 .91 .098 .19 |
      | .54 .37 .84 .92  .12 |
      | .83 .83 .63 .35  .45 |
      | .48 .23 .8  .017 .16 |

                 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 = | -4.2 -13 2.7  6.1  7.7  |
      | .96  15  -1.5 -1.7 -14  |
      | 1.5  5.4 -.83 -2.7 -2.1 |
      | .28  -2  1.1  -.17 1.2  |
      | 3.7  -9  -1.8 -1.9 13   |

                 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 = 4.05231403988182e-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 = | -4.2 -13 2.7  6.1  7.7  |
      | .96  15  -1.5 -1.7 -14  |
      | 1.5  5.4 -.83 -2.7 -2.1 |
      | .28  -2  1.1  -.17 1.2  |
      | 3.7  -9  -1.8 -1.9 13   |

                 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 :