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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 = | .62+.67i  .02+.95i  .096+.33i .03+.85i .47+.073i .63+.06i   .85+.33i 
      | .51+.09i  .42+.68i  .77+.47i  .43+.41i .73+.97i  .53+.23i   .8+.88i  
      | .76+.04i  .081+.24i .78+.03i  .23+.6i  .77+.61i  .44+.91i   .089+.11i
      | .13+.96i  .35+.6i   .23+.34i  .21+.4i  .3+.49i   .028+.096i .31+.9i  
      | .42+.041i .72+.56i  .79+.18i  .32+.97i .4+.73i   .33+.64i   .72+.3i  
      | .74+.44i  .88+.68i  .2+.67i   .6+.65i  .43+.97i  .19+.96i   .34+.75i 
      | .27+.062i .97+.22i  .36+.86i  .15+.16i .2+.26i   .88+.49i   .55+.62i 
      | 1+.2i     .55+.26i  .51+.95i  .37+.29i .93+.71i  .14+.77i   .68+.18i 
      | .89+.58i  .14+.79i  .1+.67i   .96+.31i .56+.16i  .2+.17i    .76+.18i 
      | .52+.44i  .11+.55i  .76+.12i  .67+.21i .9+.05i   .93+.31i   .1+.78i  
      -----------------------------------------------------------------------
      .053+.47i .68+.98i .63+.68i |
      .96+.72i  .52+.09i .78+.94i |
      .93+.46i  .59+.7i  .18+.56i |
      .81+.17i  .75+.13i .84+.16i |
      .95+.11i  .3+.92i  .6+.5i   |
      .35+.15i  .28+.85i .17+.16i |
      .04+.86i  .13+.79i .82+.75i |
      .16+.014i .48+.65i .6+.86i  |
      .71+.34i  .21+.12i .69+.53i |
      .38+.28i  .25+.88i .17+.3i  |

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

o14 = | .98+.61i  .81+.2i   |
      | .39+.26i  .032+.23i |
      | .54+.37i  .87+.91i  |
      | .18+.68i  .24+.24i  |
      | .05+.59i  .56+.82i  |
      | .8+.98i   .13+.73i  |
      | .6+.33i   .12+.58i  |
      | .68+.97i  .7+.29i   |
      | .24+.025i .78+.65i  |
      | .46+.28i  .61+.84i  |

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

o15 = | .27+.4i    .34-.17i   |
      | -.059-.12i -.54+.14i  |
      | -.59-.1i   -.056-.14i |
      | -.28-.1i   .19+.32i   |
      | .73-.12i   -.59+.32i  |
      | .27-.26i   .46-.35i   |
      | .27-.12i   .18+.27i   |
      | -.4-.51i   .53+.65i   |
      | .52-.04i   .5-.45i    |
      | .07+.82i   .029-.37i  |

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

o16 = 5.1178752665209e-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 = | .57 .63 .21  .45 .073 |
      | .42 .56 .29  .67 .73  |
      | .56 .7  .71  .45 .67  |
      | .84 .82 .99  .39 .14  |
      | .18 .82 .026 .77 .91  |

                 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 = | 4.4  .028 4.8  -4.3 -3.2 |
      | 1    -3.9 1.4  -.15 2    |
      | -3.4 1.3  -2.9 3.4  .8   |
      | -3.4 4.9  -7.4 4.6  1.1  |
      | 1.2  -.73 4.2  -3   -1   |

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

o20 = 9.99200722162641e-16

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

o21 = 1.19348975147204e-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 = | 4.4  .028 4.8  -4.3 -3.2 |
      | 1    -3.9 1.4  -.15 2    |
      | -3.4 1.3  -2.9 3.4  .8   |
      | -3.4 4.9  -7.4 4.6  1.1  |
      | 1.2  -.73 4.2  -3   -1   |

                 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 :