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 = | .95+.25i .44+.79i  .71+.69i .4+.41i   .26+.019i .64+.88i .16+.69i
      | .37+.33i .1+.91i   .84+.69i .58+.17i  .02+.47i  .62+.14i .18+.59i
      | .57+.86i .81+.75i  .36+.46i .76+.1i   .55+.68i  .59+.28i .85+.42i
      | .23+.2i  .75+.29i  .71+.77i .24+.92i  .017+.33i .2+.4i   .22+.77i
      | .73+.22i .79+.51i  .29+.85i .16+.74i  .1+.81i   .82+.08i .15+.56i
      | .74+.43i .7+.38i   .59+.62i .4+.19i   .57+.13i  .51+.83i .2+.58i 
      | .99+.27i .25+.022i .88+.04i .36+.045i .72+.11i  .99+.56i .06+.74i
      | .72+.1i  .16+.11i  .46+.89i .48+.67i  .39+.13i  .88+.44i .75+.72i
      | .52+.94i .03+.68i  .1+.67i  .89+.84i  .44+.1i   .2+.72i  .6+.37i 
      | .69+.68i .77+.87i  .34+.13i .19+.019i .98+.27i  .33+.68i .3+.13i 
      -----------------------------------------------------------------------
      .77+.84i .82+.18i  .06+.73i |
      .15+.55i .34+.85i  .89+.52i |
      .68+.03i .58+.26i  .53+.16i |
      .4+.68i  .93+.59i  .14+.31i |
      .47+.87i .46+.11i  .84+.76i |
      .58+.79i .58+.43i  .63+.42i |
      .14+.4i  .21+.48i  .18+.13i |
      .52+.02i .28+.032i .77+.02i |
      .11+.96i .93+.48i  .31+.98i |
      .45+.39i .66+.29i  .82+.35i |

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

o14 = | .63+.9i  .08+.25i |
      | .4+.44i  .59+.39i |
      | .38+.17i .98+.83i |
      | .37+.3i  .69+.97i |
      | .52+.58i .47+.25i |
      | .25+.56i .48+.16i |
      | .04+.86i .32+.38i |
      | .55+.95i .97+.03i |
      | .74+.58i .92+.98i |
      | .8+.73i  .54+.73i |

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

o15 = | -.36+.17i  -2.3+3i   |
      | .15-.81i   -.17+.12i |
      | .2+.2i     -2.6-2i   |
      | .05-.85i   2.4+1.8i  |
      | .47+.67i   2.3-1.8i  |
      | .41+.24i   2.2-i     |
      | .06+.42i   -2-.65i   |
      | .1-.04i    -2.1+1.4i |
      | -.084+.23i 4.1+.24i  |
      | .11-.053i  -.2+.28i  |

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

o16 = 1.54237111854025e-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 = | .56 .02 .85 .17  .41  |
      | .99 .88 .34 .19  .53  |
      | .22 .79 .18 .66  .051 |
      | .51 .87 .62 .79  .017 |
      | .39 .43 .59 .044 .55  |

                 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 = | .91  1.4   -.51 .013 -1.9 |
      | -2.4 .0045 -1.9 2    1.9  |
      | -.79 -.83  -2.5 2.4  1.5  |
      | 2.7  -.24  4.4  -2.8 -2.1 |
      | 1.9  -.066 4.2  -3.9 .22  |

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

o20 = 5.55111512312578e-16

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

o21 = 7.7715611723761e-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 = | .91  1.4   -.51 .013 -1.9 |
      | -2.4 .0045 -1.9 2    1.9  |
      | -.79 -.83  -2.5 2.4  1.5  |
      | 2.7  -.24  4.4  -2.8 -2.1 |
      | 1.9  -.066 4.2  -3.9 .22  |

                 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 :