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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 = | .87+.55i .36+.29i .88+.77i .23+.5i  .87+.98i  .17+.45i   .7+.96i 
      | .75+.38i .95+.86i .73+.81i .2+.3i   .54+.45i  .76+.56i   .12+.33i
      | .31+.82i .44+.54i .98+.91i .57+.46i .78+.81i  .86+.97i   .47+.72i
      | .43+.29i .78+.75i .65+.25i .6+.79i  .01+.79i  .94+.94i   .68+.73i
      | .5+.66i  .63+.94i .24+.84i .29+.38i .16+.31i  .08+.98i   .55+.92i
      | .27+.55i .26+.65i .3+.77i  .89+.2i  .92+.5i   .82+.82i   .5+.16i 
      | .32+.81i .51+.46i .15+.33i .56+.34i .09+.98i  .072+.014i .83+.6i 
      | .04+.71i .31+.32i .23+.23i .77+.84i .33+.33i  .5+.44i    .95+.59i
      | .67+.96i .57+.32i .93+.07i .25+.27i .033+.26i .63+.78i   .11+.1i 
      | .4+.59i  .85+.85i .28+.78i .69+.98i .014+.24i .42+.42i   .02+.95i
      -----------------------------------------------------------------------
      .32i      .02+.58i .53+.52i |
      .35+.42i  .37+.42i .82+.06i |
      .82+.15i  .24+.13i .86+.56i |
      .4+.89i   .01+.65i .26+.14i |
      .95+.88i  .3+.61i  .79+.58i |
      .19+.089i .3+i     .54+.33i |
      .88+.2i   .3+.71i  .88+.62i |
      1+.42i    .79+.82i .45+.71i |
      .59+.71i  .37+.53i .99+.11i |
      .11+.94i  .79+.53i .85+.19i |

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

o14 = | .65+.69i  .69+.65i |
      | .048+.17i .44+.97i |
      | .34+.89i  .37+.92i |
      | .08+.53i  .34+.34i |
      | .78+.72i  .49+.59i |
      | .59+.15i  .81+.84i |
      | .11+.32i  .71+.94i |
      | .36+.86i  .76+.44i |
      | .17+.68i  .41+.6i  |
      | .59+.14i  .25+.42i |

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

o15 = | 1.1+.36i  .1+.12i   |
      | .26+1.7i  .85-1.1i  |
      | -1.8-.17i 1.4+1.4i  |
      | .016+.17i .89+.65i  |
      | -.03+.93i .12-.013i |
      | 1.2-.74i  -1.5-.57i |
      | .5-.76i   -.89-.75i |
      | -.79-.09i .18+1.6i  |
      | -.55+.33i .96-1.1i  |
      | 1.4-1.2i  -1.1-.42i |

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

o16 = 1.48952049194836e-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 = | .79 .4   .7   .87  .016 |
      | .41 .22  .023 .95  .6   |
      | .83 1    .93  .083 .11  |
      | .01 .83  .42  .33  .32  |
      | .63 .099 .98  .15  .48  |

                 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 = | -.45 .94   1.5  -1.7 -.34 |
      | -.32 .17   .75  .5   -.7  |
      | .82  -1.2  -.97 .95  .99  |
      | 1.1  -.034 -.9  .58  -.17 |
      | -1.3 1.1   .2   .048 .72  |

                 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 = 3.88578058618805e-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 = | -.45 .94   1.5  -1.7 -.34 |
      | -.32 .17   .75  .5   -.7  |
      | .82  -1.2  -.97 .95  .99  |
      | 1.1  -.034 -.9  .58  -.17 |
      | -1.3 1.1   .2   .048 .72  |

                 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 :