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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 = | .14+.57i  .43+.82i .72+.79i .59+.73i .93      .33+.86i .49+.27i 
      | .2+.38i   .95+.98i .26+.24i .47+.79i .74+.11i .17+.4i  .47+.065i
      | .89+.53i  .5+.88i  .77+.25i .83+.62i .45+.86i .79+.93i .01+.53i 
      | .91+.81i  .57+.62i .95+.38i .9+.69i  .98+.53i .84+.67i .72+.45i 
      | .33+.059i .15+.97i .95+.25i .89+.74i .73+.47i .35+.44i .7+.42i  
      | .18+.28i  .99+.39i .87+.69i .85+.33i .99+.84i .88+.4i  .49+.61i 
      | .91+.3i   .03+.85i .17+.45i .38+.82i .55+.08i .64+.89i .1+.13i  
      | .4+.28i   .81+.83i .92+.32i .3+.66i  .09+.63i .35+.29i .61+.84i 
      | .44+.74i  .53+.53i .87+.74i .52+.78i .51+.12i .1+.98i  .39+.003i
      | .022+.34i .19+.44i .63+.49i .76+.23i .32+.36i .46+.4i  .98+.72i 
      -----------------------------------------------------------------------
      .81+.64i  .35+.56i  .52+.04i  |
      .13+.85i  .54+.85i  .27+.96i  |
      .23+.095i .36+.97i  .59+.05i  |
      .072+.32i .21+.044i .94+.2i   |
      .67+.1i   .61+.86i  .086+.31i |
      .41+.25i  .96+.15i  .64+.73i  |
      .91+.05i  .49+.64i  .46+.82i  |
      .06+.75i  .1+.6i    .15+.15i  |
      .92+.35i  .84+.71i  .85+.16i  |
      .62+.7i   .76+.1i   .46+.29i  |

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

o14 = | .16+.033i .34+.27i  |
      | .47+.78i  .068+.47i |
      | .51+.84i  .87+.43i  |
      | .74+.97i  .96+.37i  |
      | .35+.96i  .16+.3i   |
      | .11+.85i  .02+.64i  |
      | .64+.41i  .33+.92i  |
      | .14+.48i  .34+.64i  |
      | .03+.89i  .04+.6i   |
      | .78+.85i  .03+.69i  |

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

o15 = | .36-.046i -.3-.46i  |
      | -.54-1.1i .35-.44i  |
      | -.22+1.2i -.96+.96i |
      | .35+.13i  .03-.89i  |
      | .071-.34i .12-.37i  |
      | -.15+.47i .44+.5i   |
      | -.63-.03i .042-.39i |
      | .54-i     .57+.34i  |
      | .46+.88i  -.35+.16i |
      | .71+.11i  .8+.81i   |

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

o16 = 8.00593208497344e-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 = | .59 .31  .35 .42 .13  |
      | .6  .15  .95 .22 .036 |
      | .33 .8   .15 .65 .023 |
      | .86 .067 .3  .65 .86  |
      | .64 .21  .43 .25 .11  |

                 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 = | 1.1  -1.5 -.89 -.46 3    |
      | -9.6 -.06 3.3  .52  6.5  |
      | -1.6 1.9  .49  .26  -.85 |
      | 12   .38  -2.3 -.53 -9.6 |
      | -8.8 .55  2.2  1.9  4    |

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

o20 = 1.11022302462516e-15

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

o21 = 8.88178419700125e-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 = | 1.1  -1.5 -.89 -.46 3    |
      | -9.6 -.06 3.3  .52  6.5  |
      | -1.6 1.9  .49  .26  -.85 |
      | 12   .38  -2.3 -.53 -9.6 |
      | -8.8 .55  2.2  1.9  4    |

                 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 :