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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 = | .66+.13i .35+.94i .76+.31i .55+.58i  .87+.03i  .11+.89i  .1+.37i  
      | .74+.39i .54+.3i  .94+.61i .15+.38i  .16+.11i  .39+.66i  .11+.6i  
      | .74+.19i .64+.98i .96+.31i .55+.76i  .05+.97i  .044+.28i .41+.085i
      | .15+.43i .92+.73i .57+.23i .37+.22i  .11+.95i  .91+.58i  .64+.75i 
      | .17+.23i .52+.96i .57+.69i .14+.35i  .37+.73i  .22+.87i  .94+.3i  
      | .89+.33i .26+.78i .63+.47i .051+.46i .3+.15i   .97+.12i  .94+.09i 
      | .27+.29i .76+.37i .82+.87i .42+.95i  .55+.83i  .07+.91i  .66+.13i 
      | .96+.01i .98+.94i .48+.87i .57+.55i  .021+.11i .26+.06i  .34+.55i 
      | .45+.53i .73+.66i .72+.67i .077+.4i  .12+.8i   .31+.52i  .27+.51i 
      | .12+.64i .16+.16i .6+.9i   .36+.074i .77+.5i   .28+.63i  .45+.51i 
      -----------------------------------------------------------------------
      .93+.03i .46+.88i   .9+.02i  |
      .68+.79i .059+.088i .84+.51i |
      .2+.076i .38+.49i   .62+.01i |
      .74+.72i .5+.1i     .71+.75i |
      .57+.13i .062+.39i  .1+.66i  |
      .25+.49i .77+.18i   .7+.44i  |
      .51+.28i .6+.4i     .45+.88i |
      .97+.09i .99+.65i   .81+.83i |
      .87+.69i .3+.65i    .92+.83i |
      .4+.16i  .42+.62i   .92+.33i |

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

o14 = | .95+.36i  .1+.71i  |
      | .24+.18i  .23+.28i |
      | .25+.66i  .61+.59i |
      | .007+.34i .59+.32i |
      | .82+.85i  .59+.92i |
      | .25+.73i  .87+.66i |
      | .24+.68i  .37+.52i |
      | .05+.79i  .57+.81i |
      | .1+.59i   .93+.29i |
      | .37+.84i  .29+.95i |

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

o15 = | -.4+.96i  .54-.14i   |
      | .02-.87i  .12+.039i  |
      | .91-.45i  -.035-.35i |
      | -.76+.29i -.2+1.4i   |
      | 1.1-.05i  -.96-.54i  |
      | .37+.15i  .16-.81i   |
      | -.58-.09i .11+1.7i   |
      | -.12+.51i .57-.45i   |
      | -.27+.35i .8-.42i    |
      | -.015-.1i -.053+.39i |

                 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 = | .52 .55 .28 .27 .67 |
      | .16 .98 .73 .44 .57 |
      | .92 .91 .02 .63 .35 |
      | .76 .92 .97 .66 .49 |
      | .69 .14 .57 .45 .59 |

                 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 = | 2   -2.6 -.37  2.1  -1.3 |
      | 1.2 -.25 .0022 .94  -1.8 |
      | .32 -.69 -1.2  1.8  -.46 |
      | -5  3.6  2.2   -3.6 3.9  |
      | .88 1.1  -.11  -1.7 1.1  |

                 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 = 1.33226762955019e-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 = | 2   -2.6 -.37  2.1  -1.3 |
      | 1.2 -.25 .0022 .94  -1.8 |
      | .32 -.69 -1.2  1.8  -.46 |
      | -5  3.6  2.2   -3.6 3.9  |
      | .88 1.1  -.11  -1.7 1.1  |

                 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 :