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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 |
      | 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 = | .66+.76i .8+.97i  .74+.1i   .6+.04i  .8+.66i   .18+.26i .52+.51i
      | .87+.94i .23+.59i .56+.93i  .03+.38i .92+.01i  .52+.64i .16+.6i 
      | .95+.44i .44+.26i .087+.25i .81+.39i .81+.54i  .81+.33i .63+.12i
      | .58+.42i .8+.32i  .4+.56i   .27+.16i .57+.37i  .87+.34i .81+.69i
      | .27+.44i .57+.14i .46+.84i  .67i     .14+.56i  .42+.31i .54+.57i
      | .57+.46i .5+.84i  .061+.18i .36+.55i .78+.07i  .75+.95i .28+.64i
      | .52+.83i .65+.9i  .22+.58i  .53+.89i .058+.47i .32+.26i .87+.96i
      | .62+.6i  .74+.47i .64+.82i  .43+.12i .91+.09i  .93+.95i .53+.08i
      | .91+.22i .81+.14i .5+.42i   .3+.45i  .48+.84i  .79+.6i  .98+.79i
      | .99+.38i .28+.34i .57+.87i  .99+.99i .43+.75i  .25+.84i .1+.5i  
      -----------------------------------------------------------------------
      .5+.58i  .22+.53i .53+.75i |
      .62+.83i .96+.14i .85+.77i |
      .83+.47i .08+.98i .44+.78i |
      .68+.21i .24+.75i .84+.74i |
      .96+.65i .25+.58i .57+.33i |
      .5+.15i  .84+.61i .35+.39i |
      .8+.65i  .16+.88i .26+.33i |
      .77+.42i .23+.91i .28+.76i |
      .72+.04i .73+.68i .82+.48i |
      .64+.66i .7+.35i  .37+.26i |

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

o14 = | .33+.49i .91+.55i |
      | .74+.63i .94+.38i |
      | .62+.54i .65+.79i |
      | .22+.64i .64+.07i |
      | .89+.18i .77+.42i |
      | .13+.85i .78+.63i |
      | .9+.59i  .08+.53i |
      | .15+.68i .72+.24i |
      | .7+.06i  .63+.98i |
      | .43+.75i .96+.37i |

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

o15 = | 1.1-1.1i   -.16+1.1i |
      | -.17-.24i  1.1-.28i  |
      | -.65-.19i  -.56i     |
      | -.49+1.6i  -.42-1.3i |
      | -.54-1.2i  1.6-.29i  |
      | 1.3+.63i   -.29+.8i  |
      | -.36-.045i -.72+.05i |
      | 1-.83i     .22+.36i  |
      | .47+.098i  -.03+.54i |
      | -.54+2.1i  -.39-.83i |

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

o16 = 1.11022302462516e-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 = | .27  .9  .43  .47 .2  |
      | .49  .2  .1   .78 .14 |
      | .14  .52 .89  .48 .22 |
      | .18  .97 .052 .84 .22 |
      | .071 .89 .094 .17 .38 |

                 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 = | 3.1  1.7  -1.5 -2.4 .0043 |
      | 1.8  -.83 -.74 .28  -.35  |
      | .4   -.27 1    -.48 -.42  |
      | -1.7 .18  .75  1.6  -.56  |
      | -4.1 1.6  1.4  -.8  3.8   |

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

o20 = 2.22044604925031e-16

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

o21 = 4.44089209850063e-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 = | 3.1  1.7  -1.5 -2.4 .0043 |
      | 1.8  -.83 -.74 .28  -.35  |
      | .4   -.27 1    -.48 -.42  |
      | -1.7 .18  .75  1.6  -.56  |
      | -4.1 1.6  1.4  -.8  3.8   |

                 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 :