next | previous | forward | backward | up | top | index | toc | Macaulay2 web site

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 = | .69+.78i .85+.39i .66+.29i  .64+.01i  .039+.37i .81+.21i .25+.7i 
      | .82+.75i .62+.77i .77+.19i  .19+.12i  .37+.4i   .55+.79i .66+.09i
      | .27+.5i  .18+.61i .06+.97i  .02+.87i  .087+.44i .91+.87i .49+.48i
      | .07+.64i .55+.44i .38+.45i  .13+.5i   .67+.87i  .4+.82i  .5+.75i 
      | .94+.68i .28+.71i .65+.8i   .018+.29i .69+.52i  .96+.88i .9+.63i 
      | .68+.7i  .42+.15i .89+.41i  .57+.48i  .57+.26i  .04+.77i .12+.78i
      | .94+.45i .76+.2i  .21+.92i  .52+.69i  .3+.48i   .91+.99i .53+.92i
      | .97+.41i .54+.41i .07+.75i  .78+.79i  .72+.84i  .16+.91i .69+.02i
      | .73+.37i .24+.1i  .37+.88i  .56+.13i  .65+.39i  1+.31i   .85+.38i
      | .7+.93i  .7+.87i  .083+.23i .68+.32i  .94+.68i  .39+.67i .54+.84i
      -----------------------------------------------------------------------
      .26+.53i  .52+.67i .39+.3i   |
      .54+.98i  .17+.56i .24+.83i  |
      .6+.75i   .99+.82i .71+.93i  |
      .15+.027i .89+.27i .01+.77i  |
      .76+.3i   .74+.06i .67+.59i  |
      .28+.74i  .4+.97i  .098+.21i |
      .95+.92i  .75+.4i  .098+.44i |
      .41+.51i  .09+.58i .96+.7i   |
      .68+.41i  .89+.71i .16+.8i   |
      .28+.22i  .83+.07i .35+.046i |

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

o14 = | .03+.63i .55+i    |
      | .16+.13i .62+.84i |
      | .5+.26i  .32+.73i |
      | .13+.78i .29+.64i |
      | .88      .91+.87i |
      | .8+.86i  .56+.6i  |
      | .71+.39i .3+.19i  |
      | .43+.62i .96+.63i |
      | .91+.88i .85+.1i  |
      | .76+.82i .92+.37i |

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

o15 = | .41+.23i   1-.78i     |
      | -.43-.13i  -.4+.33i   |
      | -.09-.28i  -.094+.27i |
      | .097+.092i -.34-.21i  |
      | .65+.78i   .05-.62i   |
      | -.52+.23i  .41-.17i   |
      | .08-.57i   .37+.91i   |
      | .16-.3i    -.39+.37i  |
      | .41+.13i   .43-.57i   |
      | .18-.21i   -.14+.86i  |

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

o16 = 4.57756679852224e-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 = | .21 .48 .53  .063 .86 |
      | .28 .33 .018 .47  .74 |
      | .93 .87 .61  .17  .92 |
      | .69 .76 .53  .74  .63 |
      | .24 .74 .88  .49  .62 |

                 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 = | 6.7 -5.3 -2.9 7.7 -6.5 |
      | -18 12   10   -18 15   |
      | 9.7 -7.2 -5.1 9.1 -6.5 |
      | 2.6 -1.4 -2.6 4   -2.1 |
      | 3.7 -.83 -1.6 2.2 -2.4 |

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

o20 = 1.99840144432528e-15

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

o21 = 5.32907051820075e-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 = | 6.7 -5.3 -2.9 7.7 -6.5 |
      | -18 12   10   -18 15   |
      | 9.7 -7.2 -5.1 9.1 -6.5 |
      | 2.6 -1.4 -2.6 4   -2.1 |
      | 3.7 -.83 -1.6 2.2 -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 :