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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 = | .76+.22i  .58+.94i .48+.21i 1+.83i   .28+.48i .58+.21i .9+.9i   
      | .36+.63i  .6+.28i  .75+.8i  .86+.18i .64+.53i .6+.65i  .84+.72i 
      | .74+.87i  .28+.76i .6+.22i  .52+.52i .83+.09i .09+.57i .78+.73i 
      | .37+.92i  .66+.94i .2+.57i  .67+.31i .4+.74i  .47+.93i .39+.61i 
      | .18+.007i .9+.31i  .91+.69i .98+.67i .92+.94i .56+.83i .22+.24i 
      | 1+.79i    .93+.07i .12+.92i .59+.01i .9+.95i  .58+.04i .46+.079i
      | .2+.1i    .65+.08i .62+.14i .83+.48i .89+.7i  .23+.66i .17+.28i 
      | .73+.05i  .74+.46i .56+.79i .08+.82i .56+.03i .72+.86i .77+.54i 
      | .17+.24i  .24+.35i .27+.72i .18+.28i .75+.3i  .96+.26i .81+.99i 
      | .69+.46i  .37+.94i .1+.67i  .74+.84i .66+.33i .95+.76i .33+.23i 
      -----------------------------------------------------------------------
      .63+.75i   .02+.91i  .019+.33i |
      .14+.55i   .34+.53i  .99+i     |
      .82+.12i   .75+.16i  .51+.33i  |
      .88+.23i   .32+.062i .36+.74i  |
      .13+.13i   .62+.36i  .92+.59i  |
      .52+.27i   .23+.67i  .97+.48i  |
      .02+.0082i .49+.62i  .74+.35i  |
      .79+.79i   .23+.51i  .44+.35i  |
      .98+.45i   .48+.45i  .6+.4i    |
      .94+.83i   .38+.94i  .35+.78i  |

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

o14 = | .7+.84i  .13+.26i |
      | .42+.23i .48+.69i |
      | .47+.6i  .34+.88i |
      | .21+.59i .47+.52i |
      | .1+.84i  .72+.56i |
      | .15+.63i .02+.89i |
      | .79+.85i .86+.89i |
      | .6+.5i   .51+.07i |
      | .63+.57i .08+.97i |
      | .28+.12i .85+.16i |

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

o15 = | -.7-.17i  -.86-.42i |
      | .11+.19i  -.24-1.2i |
      | -.3-.57i  .43-1.4i  |
      | .76+.49i  .63+1.3i  |
      | .28+.79i  .11+i     |
      | -.29-1.1i .32-1.8i  |
      | .71-.26i  .42+.73i  |
      | .28+.89i  -.47+1.4i |
      | -.7-1.4i  -.37-2i   |
      | -.11+1.3i -.2+2.1i  |

                 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 = | .35  .24 .14 .12  .17  |
      | .076 .51 .1  .64  .5   |
      | .42  .97 .77 .051 .25  |
      | .17  1   .89 .88  .032 |
      | .91  .59 .44 .3   .99  |

                 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 = | 4.2  -.69 -.62 .062 -.22 |
      | 2.8  1.9  1.7  -1.2 -1.8 |
      | -4.7 -2.1 -.47 1.6  1.9  |
      | .86  .16  -1.3 .95  .072 |
      | -3.7 .41  .17  -.3  1.4  |

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

o20 = 8.88178419700125e-16

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

o21 = 6.66133814775094e-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 = | 4.2  -.69 -.62 .062 -.22 |
      | 2.8  1.9  1.7  -1.2 -1.8 |
      | -4.7 -2.1 -.47 1.6  1.9  |
      | .86  .16  -1.3 .95  .072 |
      | -3.7 .41  .17  -.3  1.4  |

                 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 :