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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 |
      | 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 = | .67+.25i  .54+.78i  .76+.65i .34+.47i  .96+.43i .5+.06i  .15+.89i 
      | .11+.27i  .95+.13i  .63+.7i  .53+.39i  .71+.16i .87+.67i .48+.19i 
      | .56+.17i  .55+.88i  .81+.7i  .13+.29i  .21+.36i .52+.59i .57+.44i 
      | .055+.29i .56+.52i  .47+.29i .19+.6i   .39+.42i .65+.41i .93+.6i  
      | .68+.88i  .61i      .57+.86i .79+.67i  .51+.24i .19+.91i .19+.3i  
      | .9+.82i   .61+.65i  .21+.14i .2+.68i   .64+.68i .65+.84i .14+.034i
      | .91+.92i  .92+.22i  .61+.51i .19+.039i .58+.06i .58+.72i .44+.98i 
      | .5+.43i   .59+.51i  .6+.63i  .11+.66i  .85+.58i .75+.71i .26+.23i 
      | .39+.49i  .11+.036i .66+.57i .91+.81i  .78+.24i .63+.35i .86+.35i 
      | .08+.53i  .4+.17i   .7+.11i  .06+.76i  .42+.37i .92+.16i .39+.31i 
      -----------------------------------------------------------------------
      .1+.63i    .27+.86i  .72+.05i  |
      .15+.57i   .64+.8i   .71+.73i  |
      .41+.11i   .72+.75i  .37+.89i  |
      .077+.39i  .71+.61i  .76+.95i  |
      .38+.24i   .19+.58i  .09+.56i  |
      .36+.35i   .55+.22i  .1+.98i   |
      .062+.088i .77+.72i  .31+.2i   |
      .29+.68i   .22+.35i  .81+.17i  |
      .77+.02i   .29+.66i  .38+.097i |
      .39+.51i   .062+.48i .43+.88i  |

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

o14 = | .97+.87i .68+.81i   |
      | .11+.38i .61+.15i   |
      | .66+.99i .43+.034i  |
      | .41+.38i .77+.36i   |
      | .76+.19i .034+.42i  |
      | .55+.13i .66+.72i   |
      | .88      .17+.41i   |
      | .96+.59i .6+.32i    |
      | .22+.39i .076+.014i |
      | .39+.66i .25+.56i   |

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

o15 = | .24-.24i  .14+.36i   |
      | -.18-.57i -.15-.44i  |
      | .97+.53i  -.67+.38i  |
      | -1.2-1.1i -.17+.088i |
      | .83+1.3i  .5+.52i    |
      | -.84-.48i -.12i      |
      | -.37+.2i  .11-.063i  |
      | -.22+.65i .4-.72i    |
      | .26-.095i .37-.13i   |
      | .79+.15i  .31-.18i   |

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

o16 = 4.23670474327502e-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 = | .3   .63 .22 .96   .35 |
      | .72  .17 .17 .95   .37 |
      | .31  .17 .45 .0029 .68 |
      | .38  .62 .87 .23   .16 |
      | .023 .24 .82 .67   .58 |

                 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 = | -.68 1.2  .35  .65  -.95 |
      | 1.7  -1.1 .47  .47  -.98 |
      | -1   .22  -.54 .79  .91  |
      | .17  .46  -.87 -.34 .7   |
      | .6   -.43 1.6  -.95 .067 |

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

o20 = 2.22044604925031e-16

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

o21 = 2.22044604925031e-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 = | -.68 1.2  .35  .65  -.95 |
      | 1.7  -1.1 .47  .47  -.98 |
      | -1   .22  -.54 .79  .91  |
      | .17  .46  -.87 -.34 .7   |
      | .6   -.43 1.6  -.95 .067 |

                 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 :