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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 = | .83+.88i .1+.81i   .38+.44i .25+.12i  .92+.29i .82+.65i .17+.61i 
      | .24+.42i .3+.23i   .33+.14i .35+.003i .79+.92i .9+.36i  .45+.77i 
      | .68+.26i .52+.04i  .44+.68i .12+.24i  .47+.8i  .36+.53i .75+.42i 
      | .96+.69i .3+.43i   .8+.7i   .38+.54i  .5+.02i  .02+.21i .54+.88i 
      | .61+.24i .48+.017i .98+.48i .24+.46i  .61+.53i .7+.32i  .64+.99i 
      | .97+.72i .1+.005i  .83+.61i .19+.95i  .15+.77i .37+.74i .025+.28i
      | .04+.73i .43+.35i  .46+.72i .65+.63i  .75+.84i .99+.76i .11+.027i
      | .17+.34i .51+.87i  .46+.57i .76+.72i  .33+.17i .36+.28i .55+.12i 
      | .31+.28i .24+.61i  .92+.24i .95+.85i  .38+.2i  1+.59i   .15+.24i 
      | .98+.54i .09+.98i  .15+.34i .022+.49i .24+.9i  .61+.29i .1+.68i  
      -----------------------------------------------------------------------
      .019+.25i .48+.72i .95+.96i   |
      .18+.89i  .76+.59i .82+.45i   |
      .42+.27i  .1+.77i  .88+.76i   |
      .18+.1i   .05+.88i .055+.035i |
      .67+.27i  .84+.22i .36+.25i   |
      .07+.54i  .93+.64i .52+.23i   |
      .95+.49i  .67+.77i .61+.57i   |
      .93+.99i  .88+.28i .26+.57i   |
      .65+.86i  .82+.28i .31+.62i   |
      .77+.51i  .86+.64i .25+.38i   |

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

o14 = | .74+.07i .11+.93i |
      | .57+.76i .99+.03i |
      | .82+i    .69+.76i |
      | .38+.73i .28+.4i  |
      | .89+.77i .75+.72i |
      | .95+.05i .56+.11i |
      | .52+.22i .23+.54i |
      | .4+.36i  .74+.5i  |
      | .93+.07i .31+.26i |
      | .53+.33i .16+.69i |

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

o15 = | .027-.071i -.6-.25i  |
      | 1.1+.72i   .15-1.9i  |
      | 1.3+3i     2.6-1.7i  |
      | -1.9-1.5i  -.45+1.5i |
      | 1.6+.45i   -.58-2i   |
      | -1.5+.47i  .71+1.3i  |
      | -1.4-.03i  1.1+1.5i  |
      | 1.2-.7i    -1.6-.22i |
      | -.24-i     -.17+i    |
      | .4-1.5i    -.19+.67i |

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

o16 = 1.22628141165784e-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 = | .58 .13   .25 .34  .88 |
      | .9  .0045 .43 .12  .11 |
      | .47 .31   .84 .8   .37 |
      | .72 .14   .81 .034 .88 |
      | .27 .19   .67 .67  .93 |

                 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 = | .99  .79  .24  -.22 -.92 |
      | 4.1  -4.8 5.1  2.2  -7.4 |
      | -2.2 .51  -.31 .85  1.4  |
      | -.1  1    -.2  -1.7 1.7  |
      | .57  -.37 -.73 .25  .64  |

                 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 = 2.22044604925031e-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 = | .99  .79  .24  -.22 -.92 |
      | 4.1  -4.8 5.1  2.2  -7.4 |
      | -2.2 .51  -.31 .85  1.4  |
      | -.1  1    -.2  -1.7 1.7  |
      | .57  -.37 -.73 .25  .64  |

                 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 :