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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 = | .75+.97i .38+.78i .96+.76i .047+.11i .73+.1i  .01+.88i .61+.84i 
      | .21+.63i .49+.33i .05+.79i .24+.34i  .3+.53i  .94+.51i .45+.33i 
      | .16+.45i .85+.77i .45+.88i .67+.53i  .2+.48i  .77+.91i .63+.77i 
      | .52+.81i .16+.3i  .85+.34i .67+.2i   .64+.56i .11+.52i .052+.5i 
      | .6+.57i  .06+.69i .35+.17i .65+.07i  .94+.61i .46+.65i .047+.31i
      | .57+.59i .06+.73i .15+.81i .72+.8i   .7+.86i  .37+.92i .57+.55i 
      | .14+.61i .94+.27i .43+.18i .39+.78i  .3+.35i  .46+.87i .66+.25i 
      | .92+.51i .9+.12i  .79+.91i .92+.74i  .07+.77i .51+.09i .83+.37i 
      | .96+.49i .44+.68i .48+.51i .09+.33i  .65+.49i .51+.28i .4+.54i  
      | .22+.54i .12+.38i .19+.65i .39+.059i .59+.99i .91+.99i .35+.6i  
      -----------------------------------------------------------------------
      .43+.94i .74+.68i .0048+.046i |
      .4+.61i  .23+.87i .74+.83i    |
      .13+.22i .59+.4i  .61+.48i    |
      .11+.11i .2+.68i  .1+.88i     |
      .42+.63i .73+.87i .78+.15i    |
      .56+.36i .53+.74i .04+.74i    |
      .5+.24i  .76+.32i .22+.38i    |
      .5+.57i  .94+.49i .34+.073i   |
      .91+.35i .33+.53i .89+.43i    |
      .42+.16i .94+.5i  .022+.18i   |

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

o14 = | .36+.45i  .94+.53i |
      | .46+.054i .18+.85i |
      | .09+.7i   .59+.15i |
      | .52+.92i  .36+.82i |
      | .03+.54i  .45+.99i |
      | .4+.68i   .35+.25i |
      | .85+.94i  .12+.96i |
      | .36+.054i .56+.66i |
      | .073+.49i .84+.52i |
      | .39+.99i  .5+.73i  |

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

o15 = | .83-.76i  .32+.79i  |
      | .97+.55i  -.65+1.6i |
      | -1        .55-.54i  |
      | -.53+.3i  -1.1i     |
      | .71-.06i  .76+.7i   |
      | -.41-.13i .47-.6i   |
      | .57+.53i  -1.1-1.7i |
      | -.97+.63i .21-.86i  |
      | .55-.18i  -.06+1.1i |
      | -.2-.51i  .6+.17i   |

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

o16 = 8.08254562088053e-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 = | .61  .36  .9   .24 .32  |
      | .099 .24  .67  .78 .71  |
      | .17  .012 .81  .49 .12  |
      | .23  .97  .089 .67 .25  |
      | .42  .2   .64  .53 .084 |

                 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 = | .31  .38  -3.6 -.97  3.9  |
      | .76  -.85 1.5  1.4   -2.3 |
      | .76  -.66 2.7  .3    -2.2 |
      | -1.5 .54  -.63 -.041 2.4  |
      | .69  1.7  -1.8 -.58  -.32 |

                 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 = 5.13478148889135e-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 = | .31  .38  -3.6 -.97  3.9  |
      | .76  -.85 1.5  1.4   -2.3 |
      | .76  -.66 2.7  .3    -2.2 |
      | -1.5 .54  -.63 -.041 2.4  |
      | .69  1.7  -1.8 -.58  -.32 |

                 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 :