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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 = | .76+.51i .34+.79i  .26+.37i .7+.72i  .98+.04i .79+.12i .99+.21i 
      | .39+.41i .31+.93i  .17+.65i .01+.68i .27+.88i .6+.14i  1+.84i   
      | .21+.18i .88+.65i  .82+.06i .04+.72i .17+.39i .61+.17i .86+.79i 
      | .34+.92i .09+.24i  .16+.32i .4+.22i  .41+.09i .4+.61i  .07+.42i 
      | .07+.44i .73+.62i  .45+.93i .97+.84i .97+.92i .87+.43i .19+.4i  
      | .88+.25i .13+.017i .7+.22i  .27+.52i .62+.61i .05+.54i .96+.7i  
      | .95+.64i .72+.19i  .75+.29i .48+.55i .77+.36i .69+.69i .32+.16i 
      | .54+.12i .49+.88i  .9+.6i   .67+.01i .1+.4i   .07+.73i .42+.092i
      | .09+.24i .31+.78i  .18+.31i .66+.52i .99+.96i .07+.64i .6+.15i  
      | .76+.06i .23+.43i  .77+.45i .97+.75i .93+.25i .92+.84i .21+.97i 
      -----------------------------------------------------------------------
      .7+.1i    .17+.93i .44+.31i |
      .27+.14i  .58+.81i .47+.24i |
      .6+.64i   .89+.24i 1+.53i   |
      1+.31i    .77+.26i .05+.42i |
      .2+.7i    .43+.55i .3+.77i  |
      .42+.015i .84+.62i .51+.33i |
      .97+.97i  .29+.19i .06+.93i |
      .17+.74i  .55+.06i .24+.52i |
      .16+.27i  .79+.28i .07+.86i |
      .41+.05i  .96+.49i .57+.64i |

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

o14 = | .55+.58i  .17+.48i    |
      | .076+.46i .17+.81i    |
      | .57+.45i  .35+.66i    |
      | .17+.8i   .29+.58i    |
      | .25+.41i  .45+.44i    |
      | .87+.5i   .72+.58i    |
      | .4+.015i  .68+.78i    |
      | .33+.97i  .26+.99i    |
      | .07+.93i  .6i         |
      | .33+.12i  .0053+.038i |

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

o15 = | -.67-2.4i  -.76-.48i  |
      | -.22+.71i  -.36+1.1i  |
      | 2.1+.99i   1.2-.73i   |
      | -.17+.007i -.48+.08i  |
      | -1.1-.19i  -.6-.29i   |
      | 1.3-.94i   .88-.53i   |
      | .51+1.9i   .76+1.4i   |
      | -1.6+1.7i  .12+.8i    |
      | -.12-.92i  -.035-.34i |
      | .9-1.1i    .34-.72i   |

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

o16 = 1.40543009465556e-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 = | .31 .57 .026 .49 .37 |
      | .84 .97 .89  .87 .11 |
      | .27 .43 .32  .13 .8  |
      | .53 .24 .49  .58 .59 |
      | .66 .3  .69  .19 .95 |

                 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 = | 3.4   -.46  -5.6 -2.7 5    |
      | .53   .67   1.2  -1.6 -.28 |
      | -3.3  .99   2.8  1.4  -2   |
      | -.5   -.097 .99  2.9  -2.4 |
      | -.048 -.59  1.2  .78  -.37 |

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

o20 = 4.44089209850063e-16

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

o21 = 6.66133814775094e-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 = | 3.4   -.46  -5.6 -2.7 5    |
      | .53   .67   1.2  -1.6 -.28 |
      | -3.3  .99   2.8  1.4  -2   |
      | -.5   -.097 .99  2.9  -2.4 |
      | -.048 -.59  1.2  .78  -.37 |

                 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 :