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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 = | .48+.054i .66+.32i  .78+.79i  .28+.2i   .64+.66i  .64+.9i   .15+.6i 
      | .83+.48i  .74+.33i  .13+.056i .93+.15i  .057+.45i .79+.22i  .46+.87i
      | .09+.75i  .1+.022i  .72+.21i  .46+.88i  .7+.47i   .64+.1i   .14+.72i
      | .1+.016i  .47+.55i  .15+.054i .43+.55i  .25+.94i  .23+.13i  .44+.16i
      | .62+.22i  .62+.31i  .049+.33i .32+.096i .56+.09i  .48+.075i .09+.63i
      | .065+.37i .57+.78i  .39+.18i  .82+.44i  .73+.98i  .87+.1i   .75+.98i
      | .04+.99i  .29+.62i  .98+.66i  .56+.57i  .26+.002i .17+.81i  .42+.37i
      | .62+.62i  .6+.54i   .67+.28i  .59+.03i  .36+.67i  .84+.55i  .76+.79i
      | .49+.87i  .48+.25i  .58+.16i  .43+.64i  .37+.93i  .04+.53i  .42+.16i
      | .9+.15i   .031+.33i .2+.91i   .84+.67i  .51+.6i   .1+.23i   .36+.34i
      -----------------------------------------------------------------------
      .98+.52i .6+.82i   .74+.39i  |
      .66+.22i .83+.05i  .63+.69i  |
      .13+.93i .96+.24i  .29+.74i  |
      .75+.91i .22+.036i .68+.73i  |
      .86+.67i .56+.78i  .65+.35i  |
      .42+.31i .46+.01i  .37+.022i |
      .16+.25i .91+.09i  .84+.18i  |
      .56+.9i  .96+.42i  .19+.92i  |
      .29+.9i  .32+.62i  .66+.89i  |
      .9+.15i  .72+.6i   .99+.26i  |

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

o14 = | .05+.68i 1+.73i   |
      | .44+.18i .75+.97i |
      | .51+.91i .92+.38i |
      | .98+.96i .56+.16i |
      | .52+.83i .86+.22i |
      | .88+.88i .59+.87i |
      | .55+.34i .23+.21i |
      | .71+.06i .21+.62i |
      | .78+.6i  .91+.07i |
      | .69+.89i .84+.96i |

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

o15 = | -.95-.19i .35+1.1i  |
      | .15+.082i .85+.32i  |
      | .15+.51i  1.2-.15i  |
      | 1.2+.36i  .63-.53i  |
      | -.54-.53i .23+.46i  |
      | -.71+.65i -.19+.84i |
      | .34+.4i   -.17-.74i |
      | .54-.58i  -.94+.12i |
      | .94-.81i  -.95-.49i |
      | -.09+.76i .74-.75i  |

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

o16 = 7.10889595793335e-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 .49 .6  .2    .71 |
      | .83 .93 .98 .19   .66 |
      | .66 .34 .33 .099  .46 |
      | .41 .49 .48 .0028 .96 |
      | .78 .26 .55 .96   .32 |

                 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 = | 6.3 -2.8 9.2 -6.5 -1.7 |
      | -60 24   -60 52   14   |
      | 55  -20  53  -48  -13  |
      | -21 7.5  -22 18   6    |
      | .54 -1.1 .41 1.2  .054 |

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

o20 = 1.06581410364015e-14

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

o21 = 7.105427357601e-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 = | 6.3 -2.8 9.2 -6.5 -1.7 |
      | -60 24   -60 52   14   |
      | 55  -20  53  -48  -13  |
      | -21 7.5  -22 18   6    |
      | .54 -1.1 .41 1.2  .054 |

                 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 :