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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 = | .66+.77i .68+.64i  .68+.74i .45+.35i  .095+.49i .6+.45i  .43+.82i  
      | .35+.4i  .32+.3i   .6+.47i  .16+.093i .63+.1i   .73+.34i .85+.56i  
      | .2+.61i  .24+.63i  .07+.66i .13+.97i  .77+.58i  .44+.19i .96+.61i  
      | .33+.86i .091+.32i .04+.62i .49+.28i  .25+.67i  .32+.77i .48+.27i  
      | .77+i    .21+.018i .25+.31i .53+.99i  .89+.52i  .8+.26i  .83+.26i  
      | .56+.11i .34+.23i  .37+.19i .89+.44i  .76+.77i  .42+.21i .59+.64i  
      | .52+.57i .23+.41i  .45+.16i .31+.87i  .2+.28i   .87+.4i  .15+.88i  
      | .89+.82i .69+.13i  .36+.21i .18+.24i  .53+.62i  .74+.57i .083+.26i 
      | .66+.99i .39+.57i  .97+.4i  .63+.48i  .078+.25i .67+.02i .097+.091i
      | .2+.24i  .83+.16i  .38+.58i .63+.18i  .82+.96i  .77+.42i .78+.57i  
      -----------------------------------------------------------------------
      .28+.36i .63+.99i  .06+.9i  |
      .85+.37i .22+.87i  .52+.71i |
      .49+.57i .13+.28i  .4+.62i  |
      .24+.26i .51+.46i  .72+.89i |
      .61+.49i .1+.45i   .65+.31i |
      .18+.61i .95+.34i  .87+.09i |
      .87+.06i .4+.62i   .01+.79i |
      .56+.92i .32+.055i .04+.7i  |
      .6+.73i  .11+.44i  .93+.66i |
      .2+.48i  .62+.19i  .6+.38i  |

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

o14 = | .41+.57i .05+.86i |
      | .61+.61i .97+.73i |
      | .67+.49i .79+.29i |
      | .5+.15i  .36+.42i |
      | .41+.47i .4+.046i |
      | .56+.33i .18+.28i |
      | .67+.32i .94+.39i |
      | .2+.064i .19+.21i |
      | .12+.5i  .09+.99i |
      | .63+.56i .08+.54i |

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

o15 = | -.39-.094i -.098-.065i |
      | -.52-.59i  -.78-1.5i   |
      | .85+.34i   1.5+1.7i    |
      | -.048+.11i -.62+.21i   |
      | 1.2-1.1i   1.9-i       |
      | -.49+.26i  -.74-.47i   |
      | -.14+.6i   .1+.46i     |
      | .51+.88i   .2+1.4i     |
      | .4-.2i     -.3-.035i   |
      | -.53-.41i  -.27-.8i    |

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

o16 = 1.00074151062168e-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 = | .18 .18 .67  .9    .23 |
      | .83 .93 .017 .45   .74 |
      | .64 .8  .56  .0043 .34 |
      | .37 .01 .093 .13   .38 |
      | .88 .87 .038 .087  .22 |

                 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 = | .22  -1.4 -.69 2    2.1  |
      | -.18 1.1  .57  -2.2 -.46 |
      | .38  -.8  1.3  .39  -.44 |
      | .98  .3   -1.1 -.44 .54  |
      | -.63 1.4  .72  .79  -2.1 |

                 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 = 4.44089209850063e-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 = | .22  -1.4 -.69 2    2.1  |
      | -.18 1.1  .57  -2.2 -.46 |
      | .38  -.8  1.3  .39  -.44 |
      | .98  .3   -1.1 -.44 .54  |
      | -.63 1.4  .72  .79  -2.1 |

                 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 :