next | previous | forward | backward | up | top | index | toc | Macaulay2 web site
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 = | .68+.99i  .23+.69i .93+.91i  .39+.95i  .93+.26i .07+.57i .67+.81i
      | .31+.23i  .93+.45i .27+.6i   .57+.87i  .13+.13i .8+.02i  .1+.054i
      | .78+.67i  .79+.58i .71+.32i  .66+.34i  .2+.37i  .92+.15i .73+.07i
      | .35+.21i  .73+.14i .05+.67i  .33+.011i .24+.85i .55+.19i .53+.55i
      | .45+.99i  .41+.88i .32+.91i  .23+.62i  .74+.01i .85+.14i .52+.05i
      | .37+.9i   .82+.32i .73+.67i  .8+.02i   .1+.57i  .9+.35i  .95+.91i
      | .48+.66i  .35+.71i .1+.91i   .19+.45i  .69+.87i .97+.46i .87+.59i
      | .095+.37i .63+.22i .31+.086i .54+.7i   .16+.58i .88+.54i .3+.75i 
      | .15+.26i  .92+.02i .91+.16i  .49+.65i  .79+.05i .68+.43i .6+.77i 
      | .8+.15i   .92+.98i .47+.68i  .12+.84i  .11+.88i .14+.78i .43+.67i
      -----------------------------------------------------------------------
      .31+.73i  .94+.5i    .37+.19i   |
      .42+.096i .13+.53i   .44+.76i   |
      .93+.54i  .08+.86i   .44+.57i   |
      .97+.57i  .13+.062i  .012+.086i |
      .087+.05i .04+.56i   .75+.16i   |
      .56+.35i  .68+.11i   .58+.12i   |
      .54+.81i  .63+.39i   .035+.28i  |
      .39+.65i  .091+.078i .009+.42i  |
      .22+.84i  .79+.28i   .13+.82i   |
      .92+.08i  .66        .11+.058i  |

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

o14 = | .93+.88i  .29+.64i |
      | .99+.56i  .21+.58i |
      | .34+.85i  .61+.33i |
      | .16+.89i  .66+.89i |
      | .012+.16i .95+.7i  |
      | .23+.52i  .68+.32i |
      | .59+.47i  .22+.54i |
      | .47+.31i  .17+.39i |
      | .39+.85i  .15+.66i |
      | .62+.77i  .15+.21i |

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

o15 = | 2.5+.2i   .8+.95i   |
      | -1.5+1.9i -.53+.61i |
      | -.03-1.1i .56-.82i  |
      | .23-.46i  -.91+.38i |
      | -.41-.11i .92i      |
      | .85-i     1.1-.48i  |
      | -.97+.1i  -.41-1.1i |
      | -.09+.52i .55-.11i  |
      | .45+.6i   -.89-.31i |
      | .19-1.4i  1.1+.55i  |

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

o16 = 7.44760245974182e-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 = | .37 .88 .44  .63 .89  |
      | .08 .19 1    .17 .2   |
      | .19 .98 .51  .48 .052 |
      | .37 .31 .089 .59 .51  |
      | .38 .38 .82  .73 .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 = | -14  21   8.2  36   -27  |
      | -.4  1.3  1.5  1.9  -2.5 |
      | -.53 1.6  .18  .72  -.63 |
      | 6.8  -13  -4.2 -18  16   |
      | 2.9  -1.9 -2   -3.9 2.4  |

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

o20 = 1.77635683940025e-15

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

o21 = 4.44089209850063e-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 = | -14  21   8.2  36   -27  |
      | -.4  1.3  1.5  1.9  -2.5 |
      | -.53 1.6  .18  .72  -.63 |
      | 6.8  -13  -4.2 -18  16   |
      | 2.9  -1.9 -2   -3.9 2.4  |

                 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 :