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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        |
      | -4.4e-16 |
      | -2.2e-15 |

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

o11 = 2.22044604925031e-15

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 = | .92+.79i  .54+.1i  .43+.44i .63+.45i  .81+.97i  .07+.99i  .93+.94i 
      | .032+.31i .71+.4i  .35+.76i .11+.17i  .16+.27i  .77+.32i  .86+.6i  
      | .81+.26i  .12+.74i .41+.85i .9+.37i   .072+.49i .01+.62i  .69+.32i 
      | .029+.45i .62+.6i  .87+.15i .73+.31i  .87+.69i  .77+.63i  .71+.52i 
      | .68+.5i   .94+.64i .68+.11i .027+.37i .37+.32i  .69+.2i   .62+.26i 
      | .28+.005i .57+.84i .58+.98i .33+.91i  .56+.83i  .72+.48i  .67+.23i 
      | .73+.11i  .53+.62i .74+.62i .62+.69i  .43+.4i   .098+.14i .4+.64i  
      | .78+.07i  .46+.73i .65+.2i  .29+.54i  .6+.32i   .98+.05i  .38+.4i  
      | .45+.14i  .37+.79i .31+.18i .66+.75i  .59+.77i  .6+.36i   .45+.049i
      | .56+.12i  .89+.14i .91+.43i .42+.99i  .13+.56i  .78+.07i  .81+.98i 
      -----------------------------------------------------------------------
      .27+.6i  .98+.36i .54+.07i  |
      .61+.44i .32+.6i  .56+.17i  |
      .48+.84i .78+.46i .2+.98i   |
      .79+.86i .52+.59i .1+.016i  |
      .93+.42i .32+.34i .55+.28i  |
      .85+.93i .2+.077i .12+.98i  |
      .59+.04i .79+.33i .12+.93i  |
      .02+.75i .14+.15i .97+.22i  |
      .74+.13i .48+.94i .074+.49i |
      .14+.87i .48+.15i .84+.18i  |

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

o14 = | .11+.27i  .81+.23i |
      | .17+.013i .74+.03i |
      | .042+.21i .31+.95i |
      | .93+.03i  .48+.19i |
      | .93+.78i  .47+.4i  |
      | .48+.78i  .92+.36i |
      | .39+.75i  .96+.78i |
      | .21+.39i  .79+.58i |
      | .88+.78i  .22+.61i |
      | .8+.34i   1+.8i    |

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

o15 = | .97-2.7i  .83+.6i   |
      | 3.2-3.4i  1.1-.02i  |
      | -.88-1.3i .19-.54i  |
      | 5.3-.49i  .31+1.6i  |
      | -3-.72i   .4-1.6i   |
      | 1.7-.69i  .01+1.1i  |
      | .14-.031i .64-.86i  |
      | -1+3.1i   -.68+.34i |
      | -3.1+3.6i -1.3      |
      | -2.3+3.1i -.71-.38i |

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

o16 = 3.28783509194982e-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 = | .95 .48 .76 .91  .18 |
      | .37 .47 .77 .33  .84 |
      | .57 .96 .75 .8   .94 |
      | .58 .82 1   .028 .34 |
      | .66 .62 .5  .92  .89 |

                 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 = | -1   -1.6 -4.3 2.7  5.2   |
      | -.71 -2   1.7  .85  -.076 |
      | 1.5  2.3  1.6  -1.3 -3.7  |
      | 1.5  .63  2.6  -2.2 -2.8  |
      | -1.1 .61  -1.5 .45  2.2   |

                 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 = 1.08246744900953e-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 = | -1   -1.6 -4.3 2.7  5.2   |
      | -.71 -2   1.7  .85  -.076 |
      | 1.5  2.3  1.6  -1.3 -3.7  |
      | 1.5  .63  2.6  -2.2 -2.8  |
      | -1.1 .61  -1.5 .45  2.2   |

                 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 :