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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 = | .34+.11i  .99+.79i .97+.28i .74+.49i .14+.26i  .43+.97i  .86+.61i
      | .09+.93i  .53+.81i .25+.83i .05+.64i .46+.22i  .35+.62i  .23+.55i
      | .42+.68i  .21+.02i .31+.84i .66+.57i .87+.26i  .13+.7i   .5+.44i 
      | .045+.33i .78+.9i  .19+.81i .17+.84i .12+.44i  .8+.81i   .45+.55i
      | .94+.63i  .16+.57i .93+.63i .16+.7i  .046+.15i .3+.76i   .57+.47i
      | .5+.66i   .06+.78i .13+.41i .17+.14i .046+.22i .73+.12i  .81+.5i 
      | .87+.19i  .93+.24i .71+.66i .47+.76i .23+.16i  .95+.96i  .72+.04i
      | .77+.26i  .47+.61i .23+.82i .82+.67i .74+.69i  .28+.92i  .6+.44i 
      | .95+.17i  .3+.005i .85+.22i .71+.47i .73+.97i  .36+.029i .59+.29i
      | .86+.13i  .72+.15i .17+.94i .41+.79i .91+.68i  .42+.21i  .55+.07i
      -----------------------------------------------------------------------
      .26+.58i  .35+.068i .37+.21i |
      .52+.75i  .57+.26i  .44+.79i |
      .1+.027i  .76+.34i  .07+.67i |
      .79+.91i  .58+.22i  .94+.63i |
      .69+.15i  .81+.62i  .63+.29i |
      .89+.6i   .47+.55i  .48+.47i |
      .93+.01i  .06+.86i  .85+.85i |
      .037+.18i .14+.7i   .75+.8i  |
      .6+.73i   .55+.86i  .52+.98i |
      .61+.11i  .76+.29i  .04+.61i |

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

o14 = | .1+.61i   .88+.01i |
      | .61+.65i  .27+.67i |
      | .76+.14i  .94+.14i |
      | .78+.84i  .95+.72i |
      | .71+.2i   .95+.15i |
      | .71+.66i  .4+.8i   |
      | .96+.42i  .16+.96i |
      | .23+.55i  .99+.1i  |
      | .23+.48i  .15+.11i |
      | .071+.21i .75+.87i |

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

o15 = | -7.6+3i   7.2+5.2i  |
      | -3.2+6.5i 8.3+.3i   |
      | 8.5+6.6i  1.3-13i   |
      | -35-12i   9.4+42i   |
      | 18-.8i    -13-18i   |
      | 9.6+6.8i  .4-14i    |
      | 27-15i    -30-17i   |
      | -17+7.7i  17+13i    |
      | -7.2+2.3i 7.8+6.5i  |
      | 9.4-1.4i  -5.8-8.4i |

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

o16 = 1.65423230669148e-14

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 = | .57 .45  .85  .37  .99   |
      | .32 .44  .84  .66  .0047 |
      | .45 .049 .087 .081 .51   |
      | .28 .71  .28  .52  .87   |
      | .56 .29  .95  .76  .72   |

                 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 = | -.39 1.7  3.4  -.49 -1.3  |
      | .82  1.6  .28  .9   -2.4  |
      | 1.6  .078 -1.5 -.95 -.044 |
      | -2.4 -.43 .019 .85  2.3   |
      | .38  -1.6 -.86 .38  1     |

                 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 = 7.7715611723761e-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 = | -.39 1.7  3.4  -.49 -1.3  |
      | .82  1.6  .28  .9   -2.4  |
      | 1.6  .078 -1.5 -.95 -.044 |
      | -2.4 -.43 .019 .85  2.3   |
      | .38  -1.6 -.86 .38  1     |

                 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 :