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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 = | .61+.15i  .17+.26i .57+.11i   .84+.98i  .12+.99i .63+.47i  .32+.95i
      | .77+.56i  .75+.1i  .27+.79i   .17+.87i  .54+.55i .34+.014i .27+.12i
      | .084+.45i .65+.91i .19+.21i   .84+.51i  .4+.17i  .04+.69i  .37+.2i 
      | .004+.45i .05+.13i .015+.052i .49+.77i  .13+.35i .18+.3i   .34+.81i
      | .27+.3i   .77+.09i .84+.46i   .84+.85i  .82+.01i .38+.21i  .11+.73i
      | .77+.77i  .75+.51i .86+.44i   .97+.32i  .97+.57i .92+.16i  .76+.53i
      | .05+.86i  .1+.6i   .81+.1i    .031+.35i .82+.13i .75+.3i   .79+.18i
      | .72+.11i  .77+i    .92+.79i   .8+.79i   .58+.85i .94+.61i  .38+.7i 
      | .066+.29i .94+.44i .3+.6i     .53+.35i  .61+.23i .45+.26i  .81+.8i 
      | .27+.73i  .76+.27i .44+.44i   .84+.95i  .21+.87i .31+.6i   .51+.42i
      -----------------------------------------------------------------------
      .7+.84i   .37+.9i  .84+.44i |
      .83+.43i  .34+.22i .49+.96i |
      .41+.49i  .35+.23i .91+.28i |
      .72+.92i  .32+.35i .27+.64i |
      .22+.8i   .26+i    .28+.33i |
      .78+.32i  .62+.1i  .25+.81i |
      .95+.12i  .95+.54i .95+.9i  |
      .49+.44i  .74+.65i .96+.07i |
      .49+.58i  .17+.4i  .99+.01i |
      .33+.089i .9+.53i  .6+.81i  |

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

o14 = | .37+.85i  .84+.54i  |
      | .72+.78i  .18+.85i  |
      | .74+.66i  .078+.44i |
      | .01+.85i  .36+.065i |
      | .44+.95i  .71+.58i  |
      | .62+.39i  .8+.18i   |
      | .69+.55i  .96+.05i  |
      | .086+.18i .17+.88i  |
      | .24+.099i .26+.83i  |
      | .81+.45i  .24+.75i  |

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

o15 = | -.23-1.9i .24-1.4i   |
      | .43-.21i  -.029+.39i |
      | .63-.08i  1.2+.86i   |
      | .68+1.7i  -.45+.65i  |
      | -.18-1.2i .81-i      |
      | -1.8+1.9i -1.5+.17i  |
      | -2-.29i   -.72-.76i  |
      | .98+.12i  .75        |
      | .49-.71i  -.071-.11i |
      | 1+.31i    .23+.88i   |

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

o16 = 9.15513359704447e-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 = | .26 .83  .93 .1  .84 |
      | .89 .69  .99 .88 .28 |
      | .64 .79  .48 .35 .82 |
      | .61 .9   .41 .4  .24 |
      | .83 .081 .58 .82 .21 |

                 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 = | 7.5  -16  -9.2 12   14   |
      | -2.1 4    2.2  -1.6 -4.1 |
      | 3.6  -4.2 -4.2 3.2  4.3  |
      | -9.4 18   11   -13  -15  |
      | -1.9 3.5  4    -4.1 -3.2 |

                 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 = 5.10702591327572e-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 = | 7.5  -16  -9.2 12   14   |
      | -2.1 4    2.2  -1.6 -4.1 |
      | 3.6  -4.2 -4.2 3.2  4.3  |
      | -9.4 18   11   -13  -15  |
      | -1.9 3.5  4    -4.1 -3.2 |

                 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 :