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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 = | .77+.22i .35+.36i 1+.92i   .031+.44i .6+.51i   .76+.31i .76+.62i 
      | .3+.34i  .51+.15i .07+.56i .45+.36i  .76+.6i   .84+.51i .86+.02i 
      | .86+.33i .85+.32i .18+.34i .06+.72i  .42+.024i .71+.22i .84+.67i 
      | .53+.57i .97+.79i .97+.63i .48+.4i   .48+.98i  .54+.7i  .9+.57i  
      | .54+.51i .68+.8i  .76+.41i .85+.69i  .96+.53i  .02+.9i  .67+.09i 
      | .38+.92i .51+.81i .65+.87i .5+.79i   .96+.91i  .21+.86i .96+.45i 
      | .13+.94i .33+.19i .65+.23i .17+.81i  .69+.05i  .35+.42i .65+.8i  
      | .81+.09i .71+.49i .67+.45i .63+.75i  .61+.3i   .24+.68i .25+.028i
      | .42+.27i .41+.78i .14+.99i .059+.32i .27+.23i  .23+.38i .34+.11i 
      | .87+.94i .22+.93i .32+.11i .95+.1i   .73+.79i  .09+.63i .46+.45i 
      -----------------------------------------------------------------------
      .85+.81i .58+.14i .64+.05i  |
      .73+.13i .76+.48i .99+.34i  |
      .46+.97i .26+.24i .95+.07i  |
      .88+.57i .4+.29i  .064+.06i |
      .8+.63i  .01+.71i .65+.27i  |
      .45+.95i .95+.64i .84+.93i  |
      .3+.72i  .79+.64i .12+.99i  |
      .97+.2i  .15+.23i .42+.46i  |
      .57+.9i  .73+.87i .33+.89i  |
      .76+.32i .11+.31i .59+.62i  |

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

o14 = | .05+.98i .14+.86i  |
      | .39i     .87+.4i   |
      | .94+.01i .14+.42i  |
      | .81+.6i  .44+.11i  |
      | 1+.6i    .087+.01i |
      | .29+.77i .52+.34i  |
      | .67+.65i .81+.73i  |
      | .88+.4i  .64+.51i  |
      | .59+.17i .78+.62i  |
      | .72+.79i .11+.18i  |

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

o15 = | .13-.39i   -1.2+.2i  |
      | .73-.94i   .21-.89i  |
      | .3+.019i   -.49-i    |
      | .24-.22i   .61-.19i  |
      | -.22+.85i  .18+1.2i  |
      | .27+.45i   .43+.26i  |
      | .29+.083i  .85-.48i  |
      | -.15+.4i   .61+.82i  |
      | -.53-.23i  -.18+.92i |
      | -.089-.26i -.28-.89i |

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

o16 = 4.96506830649455e-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 = | .23 .24 .52  .88 .13 |
      | .16 .81 .26  .56 .4  |
      | .79 .69 .016 .94 .91 |
      | .14 .4  .61  .96 .31 |
      | .89 .37 .13  1   .1  |

                 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 = | 72   9.7 8.1  -62  -17  |
      | -.58 2   -.7  -.45 .48  |
      | 100  14  11   -85  -24  |
      | -79  -11 -8.7 68   19   |
      | 17   1.5 3.4  -14  -5.4 |

                 5          5
o19 : Matrix RR    <--- RR
               53         53
i20 : norm(A*A' - I)

o20 = 2.1316282072803e-14

o20 : RR (of precision 53)
i21 : norm(A'*A - I)

o21 = 1.4210854715202e-14

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 = | 72   9.7 8.1  -62  -17  |
      | -.58 2   -.7  -.45 .48  |
      | 100  14  11   -85  -24  |
      | -79  -11 -8.7 68   19   |
      | 17   1.5 3.4  -14  -5.4 |

                 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 :