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 = | 2.2e-16  |
      | -2.2e-16 |
      | 0        |

                 3          1
o10 : Matrix RR    <--- RR
               53         53
i11 : norm oo

o11 = 2.22044604925031e-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+.04i  .93+.07i  .48+.21i .46+.27i .095+.28i .12+.11i  .082+.21i
      | .76+.67i  .35+.46i  .96+.9i  .43+.83i .69+.68i  .71+.96i  .43+.13i 
      | .77+.29i  .11+.89i  .61+.63i .4+.15i  .4+.79i   .49+.57i  .78+.95i 
      | .01+.68i  .48+.3i   .11+.15i .19+.15i .49+.65i  .28+.66i  .37+.016i
      | .26+.074i .7+.28i   .78+.29i .16+.18i .37+.67i  .94+.23i  .58+.27i 
      | .51+.97i  .72+.22i  .37+.34i .92+.12i .42+.21i  .47+.003i .09+i    
      | .32+.43i  .8+.9i    .62+.36i .25+.64i .77+.56i  .3+.38i   .67+.35i 
      | .41+.14i  .44+.15i  .7+.63i  .92+.34i .03+.93i  .4+.78i   .92+.33i 
      | .83+.14i  .48+.017i .5+.98i  .41+.54i .22+.023i .09+.8i   .33+.27i 
      | .78+.41i  .69+.83i  .57+.11i .23+.49i .88+.68i  .86+.19i  .23+.44i 
      -----------------------------------------------------------------------
      .09+.81i .94+.78i .27+.48i |
      .64+.77i .53+.97i .59+.08i |
      .67+.62i .39+.3i  .68+.86i |
      .91+.38i .62+.37i .11+.93i |
      .43+.72i .48+.91i .92+.91i |
      .23+.74i .62+.23i .91+.84i |
      .35+.19i .44+.69i .18+.29i |
      .27+.26i .85+.18i .12+.18i |
      .03+.55i .97+.11i .96+.6i  |
      .55+.49i .56+.36i .47+.66i |

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

o14 = | .86+.43i .6+.45i  |
      | .16+.53i .52+.53i |
      | .34+.12i .23+.75i |
      | 1+.97i   .27+.15i |
      | .77+.29i .7+.26i  |
      | .12+.44i .72+.84i |
      | .58+.99i .23+.35i |
      | .9+.37i  .04+.94i |
      | .68+.46i .25+.43i |
      | .36+.37i .4+.17i  |

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

o15 = | 1.6-1.4i  -.52+.16i |
      | .63+2i    .61-.34i  |
      | 5.3+.46i  -.8-.58i  |
      | -.51-2.9i -.41+2i   |
      | -2.3-1.4i .87+.35i  |
      | -3.2+2.5i .94-.81i  |
      | -.03+1.4i -.35-.55i |
      | 1.8+1.8i  -.43-.64i |
      | -2.1+1.4i .82-.46i  |
      | -.52-3.4i .05+1.1i  |

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

o16 = 1.94209711474168e-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 = | .57 .37 .21  .34 .033 |
      | .14 .8  .082 .5  .82  |
      | .77 .26 .16  .36 .31  |
      | .61 .79 .93  .35 .23  |
      | .83 .94 1    .64 .94  |

                 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 = | -3.7 .55  4.6  2.7  -2.5 |
      | -4   2.4  3.5  4.6  -4.2 |
      | 1.9  -1.5 -2.7 -1.4 2.5  |
      | 13   -2.7 -10  -8.7 7.3  |
      | -3.4 .54  2.2  .49  -.14 |

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

o20 = 1.33226762955019e-15

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

o21 = 2.44249065417534e-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 = | -3.7 .55  4.6  2.7  -2.5 |
      | -4   2.4  3.5  4.6  -4.2 |
      | 1.9  -1.5 -2.7 -1.4 2.5  |
      | 13   -2.7 -10  -8.7 7.3  |
      | -3.4 .54  2.2  .49  -.14 |

                 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 :