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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 = | .12+.55i  .54+.32i  .08+.94i  .22+.75i .8+.29i    .79+.69i .95+.36i
      | .36+.65i  .62+.19i  .99+.51i  .03+.8i  .69+.51i   .23+.79i 1+.69i  
      | .99+.24i  .87+.86i  .38+.13i  .7+.28i  .15+.92i   .53+.85i .38+.86i
      | .14+.78i  .39+.6i   .32+.47i  .98      .51+.2i    .09+.64i .04+.84i
      | .92+.34i  .53+.95i  .78+.55i  .32+.26i .17+.28i   .22+.63i .83+.98i
      | .021+.44i .49+.95i  .5+.47i   .49+.27i .66+.85i   .19+.14i .46+.81i
      | .88+.94i  .35+.006i .46+.37i  .58+.82i .2+.6i     .51+.18i .25+.74i
      | .56+.54i  .98+.85i  .86+.55i  .15+.85i .3+.33i    .23+.83i .6+.11i 
      | .26+.66i  .88+.12i  .11+.93i  .55+.13i .52+.67i   .61+.21i .92+.88i
      | .86+.51i  .35+.53i  .024+.14i .57+.78i .075+.097i .48+.49i .48+.48i
      -----------------------------------------------------------------------
      .06+.75i  .2+.72i  .024+.32i |
      .47+.088i .75+.51i .75+.15i  |
      .9+.85i   .54+.96i .16+.37i  |
      .51+.01i  .17+.26i .33+.69i  |
      .76+.16i  .21+.8i  .17+.44i  |
      .016+.15i .27+.29i .35+.074i |
      .49+.66i  .45+.75i .34+.23i  |
      .23+.17i  .45+.36i .15+.96i  |
      .34+.87i  .65+.83i .2+.26i   |
      .25+.26i  .5+.46i  .39+.26i  |

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

o14 = | .75+.23i .94+.57i |
      | .68+.26i .63+.73i |
      | .54+.68i .92+.03i |
      | .67+.16i .14+.54i |
      | .64+.27i .87+.62i |
      | .94+.36i .33+.49i |
      | .59+.89i .15+.71i |
      | .17+.36i .85+.66i |
      | .92+.43i .72+.56i |
      | .69+.58i .29+.52i |

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

o15 = | .025-.062i -.09+.8i   |
      | -.055-.19i -.39+.018i |
      | -.54-.03i  .88+.2i    |
      | .48-.22i   .2+.15i    |
      | .33-.31i   -.12+.28i  |
      | -.3+.43i   .37-.39i   |
      | .76+.12i   .65-.44i   |
      | .42-.012i  -.51-1.1i  |
      | -.52+.04i  .12+.47i   |
      | .68-.03i   -.16-.56i  |

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

o16 = 6.75322301446426e-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 = | .24 .071 .5  .65 .55 |
      | .68 .17  .23 .18 .8  |
      | .39 1    .16 .35 .87 |
      | .67 .74  .39 .42 .94 |
      | .73 .068 .51 .75 .93 |

                 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 = | -4   -2.8 -2.3 3.9  3    |
      | -.84 -1.7 -.27 2.2  .011 |
      | 3.8  1.2  -3.1 3.9  -4.3 |
      | -2.4 -3.4 1    -.74 4.1  |
      | 3    4.4  2.7  -4.7 -2.2 |

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

o20 = 1.33226762955019e-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 = | -4   -2.8 -2.3 3.9  3    |
      | -.84 -1.7 -.27 2.2  .011 |
      | 3.8  1.2  -3.1 3.9  -4.3 |
      | -2.4 -3.4 1    -.74 4.1  |
      | 3    4.4  2.7  -4.7 -2.2 |

                 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 :