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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 = | .034+.038i .34+.64i .5+.29i  .81+.13i  .75+.91i .68+.27i  .29+.73i  
      | .61+.4i    .64+.34i .67+.73i .53+.42i  .48+.92i .74+.55i  .2+.93i   
      | .16+.78i   .27+.29i .94+.36i .91+.99i  .74+.15i .46+.002i .19+.025i 
      | .16+.19i   .12+.98i .94+.8i  .72+.75i  .62+.66i .73+.05i  .89+.66i  
      | .64+.59i   .98+.37i .69+.09i .37+.22i  .28+i    .42+.049i .49+.31i  
      | .28+.57i   .08+.63i .8+.53i  .83+.96i  .77+.24i .48+.73i  .83+.41i  
      | .61+.79i   .18+.65i .68+.8i  .83+.1i   .5+.51i  .59+.39i  .059+.071i
      | .79+.62i   .69+.24i .94+.96i .24+.041i .83+.69i .76+.02i  .26+.84i  
      | .93+.33i   .87+.61i .49+.44i .34+.13i  .04+.19i .25+.013i .03+.78i  
      | .039+.06i  .89+.04i .56+.88i .2+.25i   .58+.1i  .17+.047i .03+.68i  
      -----------------------------------------------------------------------
      .18+.12i  .82+.18i  .61+.68i  |
      .76+.71i  .15+.52i  .91+.57i  |
      .91+.85i  .26+.38i  .36+.95i  |
      .83+.65i  .44+.67i  .3+.33i   |
      .62+.82i  .77+.77i  .24+.7i   |
      .018+.12i .42+.48i  .78+.02i  |
      .09+.84i  .48+.74i  .69+.8i   |
      .36+.49i  .62+.51i  .96+.42i  |
      .65+.69i  .45+.033i .28+.32i  |
      .44+.32i  .17+.053i .079+.29i |

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

o14 = | .54+.06i .31+.57i |
      | .76+.48i .77+.93i |
      | .68+.3i  .21+.83i |
      | .04+.64i .72+.65i |
      | .64+.3i  .14+.26i |
      | .39+.71i .74+.03i |
      | .53+.83i .34+.42i |
      | .4+.49i  .43+.39i |
      | .51+.73i .19+.53i |
      | .41+.23i .61+.02i |

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

o15 = | .66+1.5i  .24+.33i   |
      | 1.2-.57i  .24-.51i   |
      | -.16+.92i -.061+.25i |
      | -.87-.68i .072+.49i  |
      | .39+1.6i  1.2+.26i   |
      | 1.4-.38i  -.45-.52i  |
      | -.12+.11i .12-.26i   |
      | -.62+.38i .36+.2i    |
      | -.64-1.2i -.97-.8i   |
      | .38-1.7i  -.11+.06i  |

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

o16 = 1.24126707662364e-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 = | .33 .9  .13 .28 .26  |
      | .93 .77 .96 .72 .14  |
      | .88 .61 .69 .15 .77  |
      | .54 .28 .54 .28 .053 |
      | .74 .12 .24 .51 .66  |

                 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 = | 2.2  -5.6 -1.5 11   1.2  |
      | 1.2  -.21 .061 .23  -.53 |
      | -2.4 3.9  1.7  -6.2 -1.4 |
      | -.77 3.1  -.66 -4.9 .83  |
      | -1.2 2.4  1.6  -6.1 .14  |

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

o20 = 1.77635683940025e-15

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

o21 = 1.72084568816899e-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 = | 2.2  -5.6 -1.5 11   1.2  |
      | 1.2  -.21 .061 .23  -.53 |
      | -2.4 3.9  1.7  -6.2 -1.4 |
      | -.77 3.1  -.66 -4.9 .83  |
      | -1.2 2.4  1.6  -6.1 .14  |

                 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 :