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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 = | .44+.5i  .07+.52i  .75+.83i   .66+.33i  .17+.26i  .71+.15i  .83+.11i 
      | .95+.98i .27+.008i .61+.73i   .32+.031i .24+.039i .4+.32i   .62+.65i 
      | .65+.63i .72+.47i  .36+.63i   .45+.76i  .044+.45i .068+.45i .51+.23i 
      | .06+.87i .12+.45i  .53+.09i   .87       .28+.91i  .27+.63i  .11+.34i 
      | .92+.7i  .71+.66i  .17+.69i   .29+.99i  .86+.93i  .41+.011i .051+.38i
      | .95+.57i .66+.09i  .33+.61i   .69+.02i  .69+.41i  .76+.78i  .59+.67i 
      | .41+.8i  .21+.82i  .99+.26i   .75+.13i  .81+.03i  .07+.71i  .7+.79i  
      | .57+.73i .29+.2i   .05+.6i    .68+.89i  .8+.16i   .026+.47i .38+.83i 
      | .75+.03i .99+.64i  .046+.016i .69+.33i  .75+.68i  .81+.43i  .31+.82i 
      | .38+.5i  .39+.23i  .83+.22i   .84+.93i  .15+.71i  .45+.31i  .95+.24i 
      -----------------------------------------------------------------------
      .01+.74i  .28+.81i   .25+.8i  |
      .48+.13i  .25+.9i    .99+.86i |
      .45+.76i  .6+.74i    .97+.01i |
      .95+.2i   .99+.25i   .39+.47i |
      .28+.25i  .45+.95i   .37+.14i |
      .75+.24i  .69+.37i   .28+.67i |
      .13+.051i .94+.1i    .5+.96i  |
      .056+.24i .59+.2i    .55+.48i |
      .96+.2i   .73+.07i   .27+.95i |
      .75+.74i  .078+.009i .65+.54i |

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

o14 = | .68+.73i  .87+.41i |
      | .78+.39i  .4+.21i  |
      | .76+.8i   .64+.53i |
      | .21+.32i  .76+.04i |
      | .37+.22i  .45+.99i |
      | .64+.98i  .13+.31i |
      | .43+.9i   .73+.04i |
      | .12+.097i .91+.77i |
      | .41+.69i  .9+.65i  |
      | .58+.26i  .75+.25i |

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

o15 = | 1.3-2.1i  .87-1.8i  |
      | 1.3-.79i  .61-1.2i  |
      | -1.3-.95i -1.4-.52i |
      | -.21-2.4i 1.5-1.7i  |
      | -2.8+2i   -2.8+1.5i |
      | -.97+.78i -.51+2.1i |
      | 2.2+3.5i  1.6+.99i  |
      | 2+3.5i    1.2+2.4i  |
      | 1.9+.53i  1.7+.57i  |
      | -2.6-1.9i -1.9-.79i |

                 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 = | .44 .87 .37  .7   .27 |
      | .99 .48 .075 .63  .61 |
      | .91 .33 .033 .28  .28 |
      | .14 .45 .73  .049 .04 |
      | .53 .47 .15  .91  .67 |

                 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 = | -.73   -3.7 4.4  .18  1.8  |
      | 2.7    8.8  -7.3 -.73 -5.9 |
      | -1.5   -4.7 3.7  1.8  3.2  |
      | -.73   -11  8.5  -.3  6.7  |
      | -.0072 13   -11  .36  -5.6 |

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

o20 = 8.88178419700125e-16

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

o21 = 2.42254133420161e-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 = | -.73   -3.7 4.4  .18  1.8  |
      | 2.7    8.8  -7.3 -.73 -5.9 |
      | -1.5   -4.7 3.7  1.8  3.2  |
      | -.73   -11  8.5  -.3  6.7  |
      | -.0072 13   -11  .36  -5.6 |

                 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 :