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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 = | 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 = | .16+.27i .38+.89i .92+.05i .1+.8i   .06+.65i .78+.84i  .22+.74i 
      | .59+.73i .86+.79i .71+.35i .36+.41i .07+.74i .32+.25i  .29+.072i
      | .73+.72i .84+.08i .34+.1i  .97+.2i  .7+.05i  .68+.4i   .63+.31i 
      | .44+.24i .65+.5i  .46+.5i  .49+.17i .26+.27i .92+.76i  .34+.82i 
      | .96+.35i .87+.19i .14+.98i .7+.55i  .66+.06i .55+.22i  .54+.9i  
      | .4+.55i  .48+.94i .11+.85i .69+.35i .72+.19i .76+.88i  .48+.35i 
      | .63+.13i .31+.44i .27+.52i .32+.52i .55+.14i .28+.77i  .44+.17i 
      | .21+.33i .72+.95i .85+.6i  .75+.54i .03+.8i  .84+.01i  .41+.8i  
      | .06+.67i .61+.12i .18+.38i .8+.89i  .7+.28i  .47+.054i .53+.55i 
      | .07+.99i .79+.39i .12+.46i .93+.54i .03+.7i  .96+.64i  .11+.17i 
      -----------------------------------------------------------------------
      .052+.44i .21+.86i  .31+.14i |
      .56+.09i  .58+.73i  .91+.1i  |
      .22+.21i  .41+.9i   .29+.62i |
      .044+.29i .027+.23i .22+.23i |
      .35+.36i  .84+.44i  .73+.05i |
      .79+.64i  .78+.58i  .34+.66i |
      .52+.25i  .25+.71i  .25+.76i |
      .57+.04i  .56+.14i  .85+.8i  |
      .14+.66i  .28+.61i  .89+.53i |
      .77+.46i  .19+.79i  .57+.76i |

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

o14 = | .71+.65i  .54+.3i   |
      | .38+.88i  .66+.84i  |
      | .6+.94i   .87+.68i  |
      | .8+.52i   .28+.15i  |
      | .72+.41i  .46+.47i  |
      | .57+.8i   .59+.49i  |
      | .19+.94i  .15+.78i  |
      | .055+.14i .25+.095i |
      | .052+.13i .4+.63i   |
      | .99+.35i  .21+.66i  |

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

o15 = | -.53+.03i .28+.2i    |
      | .76+.98i  .25-.38i   |
      | -.2+.14i  .95-.32i   |
      | -.6+.43i  .77+.25i   |
      | -.57-.01i .37+.2i    |
      | .43-1.1i  .001+.41i  |
      | .21-.4i   -1.2+.76i  |
      | .21+.049i -.97-.83i  |
      | .86-.26i  .51-.36i   |
      | .18+.091i -.34-.007i |

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

o16 = 8.89911452410874e-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 = | .79 .24 .3  .67  .46 |
      | .54 .33 .22 .92  .1  |
      | .19 .55 .25 .4   .63 |
      | .36 .59 .68 .092 .26 |
      | .74 .45 .14 .12  .84 |

                 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 = | -.57 1    -2   .3   1.6  |
      | -4.2 2.9  -.55 .63  2.2  |
      | 3    -2.3 .68  1.1  -2.2 |
      | .9   .21  .98  -.59 -1.1 |
      | 2.1  -2.1 1.8  -.71 -.83 |

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

o20 = 5.68989300120393e-16

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

o21 = 4.44089209850063e-16

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 = | -.57 1    -2   .3   1.6  |
      | -4.2 2.9  -.55 .63  2.2  |
      | 3    -2.3 .68  1.1  -2.2 |
      | .9   .21  .98  -.59 -1.1 |
      | 2.1  -2.1 1.8  -.71 -.83 |

                 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 :