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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 = | .24+.37i  .93+.75i  .11+.37i .5+.41i   .13+.23i .88+.86i .24+.087i
      | .82+.6i   .26+.067i .38+.97i .85+.74i  1+.83i   .02+.91i .2+.23i  
      | .9+.74i   .067+.44i .9+.07i  .27+.67i  .28+.68i .49+.13i .27+.092i
      | .83+.06i  .07+.53i  .17+.29i .92+.22i  .39+.91i .2+.045i .73+.45i 
      | .72+.66i  .69+.12i  .83      .38+.045i .56+.61i .83+.83i .29+i    
      | .47+.1i   .49+.93i  .43+.32i .28+.2i   .071+.4i .18+.76i .48+.25i 
      | .06+.94i  .8+.88i   .75+.08i .8+.19i   .92+.64i .76+i    .57+.57i 
      | .54+.71i  .93+.78i  .99+.53i .56+.77i  .34+.31i .43+.99i .01+.55i 
      | .25+.027i .05+.2i   .33+.51i .91+.4i   .36+.36i .37+.79i .55+.47i 
      | .98+.83i  .98+.94i  .34+.83i .8+.21i   .1+.73i  .74+.17i .8       
      -----------------------------------------------------------------------
      .64+.91i   .94+.63i .19+.15i |
      .71+.82i   .97+.97i 1+.79i   |
      .044+.018i .76+.63i .43+.56i |
      .55+.2i    .44+.51i .38+.92i |
      .92+.43i   .6+.09i  .97+.21i |
      .78+.12i   .75+.2i  .47+.82i |
      .39+.89i   .57+.36i .19+.82i |
      .26+.81i   .3+.58i  .25+.93i |
      .7+.25i    .36+.6i  .2+.91i  |
      .28+.004i  .68+.18i .12+.55i |

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

o14 = | .51+.86i .36+.38i |
      | .04+.98i .15+.82i |
      | .013+.4i .61+.04i |
      | .43+.35i .6+.59i  |
      | .19+.32i .42+.68i |
      | .26+.24i .36+.42i |
      | .42+.15i .81+.2i  |
      | .61+.89i .96+.95i |
      | .11+.28i .72+.97i |
      | .25+.54i .01+.54i |

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

o15 = | -.73-.06i -.95+.65i  |
      | .13-.76i  -.13-.3i   |
      | 1.4+.2i   1.3-.36i   |
      | 1-.71i    -.07-.7i   |
      | -.43-.35i -.046+.35i |
      | -.74+1.1i .43+1.2i   |
      | .21+.91i  .83-.22i   |
      | 1.8-.78i  .18-.73i   |
      | -1.1+.05i -.31-.11i  |
      | .2+1.1i   .64-.08i   |

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

o16 = 5.55111512312578e-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 = | .53  .34  .36 .56 .12 |
      | .97  .26  .14 .16 .39 |
      | .73  .19  .54 .75 .64 |
      | .023 .51  .43 .69 .7  |
      | .78  .081 .63 .55 .41 |

                 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 = | .33  .74  .4  -.71 -.21 |
      | 1.2  1.2  -3  1.6  .52  |
      | -.54 -.31 -4  1.3  4.5  |
      | 1.5  -1.3 3.9 -1.3 -3   |
      | -2.1 .64  .83 .79  -.21 |

                 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 = 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 = | .33  .74  .4  -.71 -.21 |
      | 1.2  1.2  -3  1.6  .52  |
      | -.54 -.31 -4  1.3  4.5  |
      | 1.5  -1.3 3.9 -1.3 -3   |
      | -2.1 .64  .83 .79  -.21 |

                 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 :