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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 = | 1+.26i   .41+.26i  .21+.25i  .38+.34i .79       .36+.99i .26+.92i
      | .88+.95i .58+.57i  .48+.15i  .64+.03i .08+.94i  .1+.83i  .5+.36i 
      | .44+.91i .095+.21i .93+.25i  .37+.56i .035+.18i .28+.73i .2+.11i 
      | .27+.52i .91+.45i  .47+.56i  .81+.96i .68+.84i  .81+.73i .11+.81i
      | .17+.46i .3+.23i   .016+.37i 1+.44i   .21+.79i  .23+.84i .35+.24i
      | .82+.14i .84+.45i  .98+.59i  .1+.55i  .15+.13i  .73+.99i .37+.16i
      | .74+.39i .52+.34i  .49+.87i  .87+.34i .28+.44i  .29+.22i .81+.34i
      | .81+.54i .15+.7i   .32+.25i  .33+.68i .19+.46i  .65+.66i .81+.07i
      | .81+.59i .69+.86i  .24+.13i  .59+.46i .9+.28i   .12+.88i .47+.12i
      | .49+.33i .4+.26i   .77+.03i  .33+.95i .86+.05i  .89+.9i  .34+.84i
      -----------------------------------------------------------------------
      .13+.038i .69+.35i  .62+.51i |
      .26+.21i  .042+.1i  .32+.4i  |
      .55+.38i  .06+.75i  .66+.66i |
      .51+.96i  .28+.053i .85+.45i |
      .35+.48i  .53+.38i  .47+.15i |
      .71+.05i  .9+.97i   .53i     |
      .45+.52i  .093+.34i .83+.95i |
      .96+.32i  .38+.8i   .92+.44i |
      .06+.63i  .95+.07i  .42+.79i |
      .49+.58i  .41+.47i  .88+.03i |

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

o14 = | .28+.056i .56+.02i  |
      | .32+.18i  .77+.55i  |
      | .68+.63i  .86+.87i  |
      | .27+.85i  .84+.98i  |
      | .57+.11i  .98+.22i  |
      | .24+.29i  .19+.081i |
      | .6+.47i   .82+.96i  |
      | .49+.11i  .24+.44i  |
      | .24+.82i  .72+.07i  |
      | .25+.71i  .81+.61i  |

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

o15 = | .58+.12i    -.33+.15i  |
      | -.28+.44i   .34-.42i   |
      | .043+.18i   .3+.65i    |
      | .63-.14i    1.3+.64i   |
      | -.25+.18i   .2+.18i    |
      | -.32-.073i  .16-.07i   |
      | -.26+.093i  -.27-.088i |
      | .57-.43i    -.68-.23i  |
      | .012+.016i  -.46-1.1i  |
      | -.066-.099i .28-.067i  |

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

o16 = 8.48363141713211e-16

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 = | .19  .045   .048 .25 .047 |
      | .098 .29    .89  .18 .081 |
      | .52  .21    .97  .72 .86  |
      | .052 .096   .096 .43 .16  |
      | .88  .00011 .85  .32 .56  |

                 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 = | 8.8  -.34 1.8  -7   -1.4 |
      | 28   .23  10   -26  -11  |
      | -9   1.2  -3.2 7.8  3.3  |
      | -6.9 .26  -3.4 9.8  2.9  |
      | 3.7  -1.4 4    -6.4 -2.6 |

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

o20 = 1.55431223447522e-15

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

o21 = 4.88498130835069e-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 = | 8.8  -.34 1.8  -7   -1.4 |
      | 28   .23  10   -26  -11  |
      | -9   1.2  -3.2 7.8  3.3  |
      | -6.9 .26  -3.4 9.8  2.9  |
      | 3.7  -1.4 4    -6.4 -2.6 |

                 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 :