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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 = | .6+.78i  .77+.78i  .78+.13i .73+.04i .22+.18i .9+.9i    .24+.23i  
      | .23+.35i .28+.61i  .59+.99i .33+.41i .49+.21i .18+.14i  .43+.71i  
      | .42+.21i .42+.95i  .53+.07i .13+.05i .41+.52i .05+.55i  .88+.56i  
      | .47+.47i .49+.27i  .48+.13i .54+.47i .96+.76i .92+.18i  .17+.41i  
      | .81+.18i .53+.99i  .69+.75i .03+.8i  .34+.76i .83+.78i  .87+.59i  
      | .33+.69i .91+.74i  .44+.72i .08+.59i .22+.36i .094+.31i .09+.95i  
      | .19+.1i  .38+.002i .31+.32i .18+.25i .45+.59i .8+.57i   .96+.45i  
      | .44+.42i .65+.04i  .15+.27i .84+.11i .12+.42i .42+.1i   .097+.012i
      | .03+.66i .68+.38i  .56+.23i .33+.23i .43+.16i .17+.51i  .1+.68i   
      | .6+.19i  .48+.71i  .18+.59i .26+.81i .19+.28i .49+.53i  .26+.61i  
      -----------------------------------------------------------------------
      .75+.49i .22+.45i  .77+.39i  |
      .93+.98i .54+.3i   .3+.44i   |
      .5+.48i  .31+.57i  .49+.22i  |
      .89+.8i  .34+.085i .74+.88i  |
      .3+.72i  .17+.069i .054+.38i |
      .67+.57i .88+.71i  .49+.45i  |
      .95+.19i .71+.63i  .85+.92i  |
      .07+.75i .93+.53i  .67+.13i  |
      .88+.38i .56+.68i  .08+.76i  |
      .7+.77i  .88+.76i  .06+.6i   |

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

o14 = | .38+.93i   .77+.81i    |
      | .4+.97i    .75+.27i    |
      | .88+.2i    .34+.72i    |
      | .26+.79i   .09+.83i    |
      | .085+.032i .87+.92i    |
      | .83+.8i    .23+.19i    |
      | .08+.79i   .36+.66i    |
      | .015+.39i  .62+.74i    |
      | .23+.78i   .42+.17i    |
      | .66+.63i   .0076+.024i |

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

o15 = | -.59-.33i .97+.04i  |
      | -.32-.75i .32+.35i  |
      | -.12+1.6i .85-1.8i  |
      | -.11-.85i -.3+1.5i  |
      | -1+.16i   1.4+.37i  |
      | .29+.59i  .14-.11i  |
      | .43-.91i  -.06+.74i |
      | .85-.49i  -.85+.35i |
      | .61+.44i  -.17-.84i |
      | .32+.93i  -.38-.29i |

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

o16 = 1.51006657275581e-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 = | .76  .19 .73 .49  .91 |
      | .75  .19 .29 .54  .74 |
      | .37  .51 .56 .73  .5  |
      | .092 .43 .85 .09  .72 |
      | .57  .11 .86 .071 .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 = | -5.2 3.6   .64  -.99 3.7  |
      | -8.5 5.4   1.3  1.4  3.2  |
      | 1.8  -2.7  .81  -.73 .77  |
      | 5.2  -3.5  .84  -1   -2.7 |
      | 3    -.055 -1.9 1.7  -2.9 |

                 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 = 1.77635683940025e-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 = | -5.2 3.6   .64  -.99 3.7  |
      | -8.5 5.4   1.3  1.4  3.2  |
      | 1.8  -2.7  .81  -.73 .77  |
      | 5.2  -3.5  .84  -1   -2.7 |
      | 3    -.055 -1.9 1.7  -2.9 |

                 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 :