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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 = | .45+.92i .74+.26i .88+.53i  .49+.96i .97+.38i  .53+.69i .03+.33i
      | .28+.22i .51+.59i .46+.85i  .42+.37i .19+.049i .39+.59i .51+.79i
      | .78+.03i .65+.43i .18+.84i  .31+.17i .25+.26i  .16+.61i .56+.74i
      | .63+i    .85      .076+.45i .56+.33i .89+.25i  .21+.89i .89+.47i
      | .86+.3i  .26+.34i .7+.39i   .77+.08i .75+.06i  .68+.51i .97+.07i
      | .69+.5i  .26+.89i .81+.72i  .91+.3i  .28+.51i  .52+.46i .68+.99i
      | .13+.32i .94+.64i .64+.68i  .22+.36i .77+.05i  .89+.23i .6+.2i  
      | .82+.53i .41+.16i .74+.22i  .86+.04i .39+.86i  .04+.91i .53+.76i
      | .68+.73i .91+.33i .47+.21i  .86+.12i .78+.24i  .46+.93i .36+.56i
      | .21+.45i .47+.12i .55+.42i  .86+.02i .7+.79i   .43+.28i .72+.36i
      -----------------------------------------------------------------------
      .44+.87i .45+.59i  .53+.16i |
      .34+.54i .08+.24i  .61+.21i |
      .13+.51i .45+.69i  .77+.3i  |
      .73+.08i .15+.43i  .56+.18i |
      .1+.79i  .67+.91i  .48+.79i |
      .31+.79i .78+.73i  .28+.19i |
      .23+.61i .78+.6i   .17+.15i |
      .89+.25i .86+.13i  .22+.52i |
      .91+.28i .08+.49i  .58+.49i |
      .62+.47i .098+.15i .28+.13i |

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

o14 = | .11+.52i  .62+.72i  |
      | .93+.91i  .35+.22i  |
      | .84+.64i  .94+.61i  |
      | .13+.58i  .76+.89i  |
      | .43+.8i   .95+.07i  |
      | .009+.35i .8+.55i   |
      | .65+.01i  .08+.88i  |
      | .024+.33i .95+.32i  |
      | .98+.73i  .02+.55i  |
      | .19+.75i  .34+.091i |

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

o15 = | -.23+2i   .73-.28i  |
      | -.08-.55i .61i      |
      | .52-.68i  .19-.13i  |
      | -.05-1.5i .11-.033i |
      | .072+.05i .1+.078i  |
      | -.15+.79i -.53-.05i |
      | .017+.36i .45-.23i  |
      | 2.2+1.7i  -.83-.39i |
      | -.76-1.5i .48+.53i  |
      | .34-1.1i  .24+.25i  |

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

o16 = 1.09344286978131e-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 = | .85 .55 .44 .89 .95  |
      | .38 .79 .7  .1  .47  |
      | .16 .67 .55 .55 .054 |
      | .57 .75 .23 .77 .33  |
      | .51 .94 .93 .18 .83  |

                 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 = | 3.3  12   -.88 -2.4 -9.7 |
      | -2.7 -4.3 -.96 3.4  4.2  |
      | 2.3  5.5  2    -3.8 -4.4 |
      | .22  -3   1.4  .14  1.3  |
      | -1.6 -8.3 -.85 1.8  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 = 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 = | 3.3  12   -.88 -2.4 -9.7 |
      | -2.7 -4.3 -.96 3.4  4.2  |
      | 2.3  5.5  2    -3.8 -4.4 |
      | .22  -3   1.4  .14  1.3  |
      | -1.6 -8.3 -.85 1.8  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 :