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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 = | .72+.56i  .78+.17i    .51+.75i   .02+.89i  .58+.94i  .39+.99i  
      | .24+.42i  .12+.25i    .61+.87i   .026+.08i .68+.29i  .4+.11i   
      | .45+.5i   .69+.76i    .69+.04i   .42+.85i  .38+.98i  .94+.83i  
      | .23+.31i  .12+.034i   .62+.64i   .95+i     .66+.96i  .29+.28i  
      | .21+.9i   .1+.12i     .14+.94i   .02+.71i  .29+.027i .17+.8i   
      | .24+.37i  .014+.0051i .94+.8i    .75+.66i  .59+.8i   .61+.11i  
      | .79+.5i   .19+.16i    .18+.036i  .25+.55i  .015+.13i .87+.77i  
      | .034+.18i .46+.1i     .88+.68i   .57+.29i  .04+.73i  .89+.59i  
      | .92+.18i  .15+.43i    .68+.3i    .82+.6i   .54+.6i   .037+.027i
      | .51+.03i  .35+.86i    .059+.093i .74+.36i  .39+.99i  .41+.42i  
      -----------------------------------------------------------------------
      .85+.75i  .87+.89i  .55+.42i  .21+.46i |
      .69+.52i  .89+.27i  .79+.18i  .93+.42i |
      .41+.35i  .72+.27i  .005+.44i .37+.5i  |
      .92+.81i  .22+.42i  .94+.29i  .2+.5i   |
      .72+.36i  .31+.74i  .06+.5i   .88+.3i  |
      .69+.58i  .73       .06+.97i  .71+.25i |
      .2+.67i   .47+.96i  .7+.28i   .85+.31i |
      .34+.77i  .27+.46i  .7+.19i   .27+.3i  |
      .063+.17i .36+.94i  .86+.17i  .29+.16i |
      .74+.71i  .061+.45i .68+.46i  .62+.91i |

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

o14 = | .71+.75i  .26+.022i |
      | .79+.56i  .86+.41i  |
      | .8+.75i   .96+.75i  |
      | .93+.94i  .96+.93i  |
      | .26+.22i  .93+.44i  |
      | .45+.83i  .49+.95i  |
      | .47+.65i  .69+.01i  |
      | .22+.66i  .88+.46i  |
      | .27+.067i .58+.68i  |
      | .29+.29i  .62+.7i   |

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

o15 = | 1.1-.99i  -1.7-1.3i |
      | -1.4-.33i 1.6-.67i  |
      | -.85-.55i .19+.57i  |
      | -.19+.36i 1.3-.45i  |
      | 2.3-.84i  -1.3+1.4i |
      | 1.1-i     -.53+.95i |
      | -.63-.74i .43+.036i |
      | -.71+1.2i 1.1-.37i  |
      | -.03+1.2i .73-.34i  |
      | -.38+1.2i -.51i     |

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

o16 = 1.24126707662364e-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 = | .014 .34 .37 .52 .53 |
      | .34  .89 .59 .97 .92 |
      | .56  .34 .49 .12 .31 |
      | .52  .18 .58 .8  .27 |
      | .41  .85 .73 .3  .93 |

                 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 = | 9.1  2.9  15   -7.6 -11  |
      | -17  -.59 -18  9.4  14   |
      | -12  -5   -18  11   15   |
      | -.25 .9   -.66 .71  -.73 |
      | 21   2.9  24   -14  -18  |

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

o20 = 3.5527136788005e-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 = | 9.1  2.9  15   -7.6 -11  |
      | -17  -.59 -18  9.4  14   |
      | -12  -5   -18  11   15   |
      | -.25 .9   -.66 .71  -.73 |
      | 21   2.9  24   -14  -18  |

                 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 :