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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 = | .77+.69i .77+.01i  .93+.07i  .01+.9i   .1+.99i   .27+.39i   .24+i    
      | .12+.64i .69+.85i  .057+.08i .45+.006i .55+.58i  .98+.26i   .72+.86i 
      | .31+.86i .025+.41i .69+.26i  .62+.07i  .26+.43i  .49+.29i   .13+.28i 
      | .88+.38i .52+.04i  .8+.98i   .51+.72i  .66+.81i  .66+.1i    .018+.34i
      | .09+.83i .16+.8i   .82+.4i   .41+.41i  .74+.82i  .62+.52i   .8+.68i  
      | .28+.58i .37+.16i  .67+.58i  .04+.7i   .4+.048i  .69+.42i   .68+.02i 
      | .26+.5i  .45+.74i  .52+.19i  .39+.46i  .66+.6i   .41+.38i   .33+.36i 
      | .74+.5i  .86+.85i  .3+.036i  .7+.01i   .42+.34i  .059+.044i .7+.64i  
      | .93+.03i .19+.96i  .1+.17i   .12+.74i  .81+.6i   .68+.14i   .89+.1i  
      | .02+.55i .21+.72i  .47+.22i  .63+.3i   .07+.047i .97+.31i   .38+.95i 
      -----------------------------------------------------------------------
      .51+.12i  .66+.36i  .2+.2i   |
      .18+.44i  .17+.25i  .39+.45i |
      .34+.009i .46+.49i  .83+.5i  |
      .79+.78i  .16+.032i .54+.16i |
      .58+.05i  .22+.92i  .02+.97i |
      .12+.76i  .83+.11i  .51+.5i  |
      .34+.75i  .32+.96i  .49+.97i |
      .71+.49i  .83+.78i  .23+.68i |
      .18+.069i .87+.23i  .07+.97i |
      .97+.62i  .56+.4i   .66+.83i |

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

o14 = | .25+.1i  .96+.24i |
      | .49+.97i .74+.51i |
      | .11+.89i .21+.32i |
      | .46+.04i .89+.72i |
      | .53+.78i .02+.59i |
      | .79+.92i .29+.48i |
      | .77+.71i .11+.94i |
      | .07+.75i .28+.79i |
      | .86+.99i .69+.37i |
      | .8+.8i   .5+.62i  |

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

o15 = | -.8-7.5i  -3.8+6.3i |
      | -4-1.9i   2.7+3.4i  |
      | .14+3.6i  1.8-2.3i  |
      | -7.7-.18i 4.7+3.7i  |
      | 9.3+1.6i  -5.7-6.5i |
      | -11+3i    11+4.1i   |
      | .84-4.5i  -3.6+3.2i |
      | 4.3-2.7i  -3.7-.07i |
      | -2.7+11i  8.3-6.8i  |
      | 5.3-5.3i  -6.6+1.3i |

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

o16 = 4.92768959440774e-15

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 = | .35  .71 .41 .95 .87 |
      | .19  .3  .34 .46 .59 |
      | .88  .88 .61 .16 .86 |
      | .033 .24 .42 .24 .91 |
      | .44  .64 .04 .29 .39 |

                 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 = | -10  21  -2   -4.2 6    |
      | 11   -22 2.8  3.4  -5.1 |
      | 7.9  -13 3.4  1.2  -7.6 |
      | -1.3 5.5 -1.1 -1.8 .89  |
      | -5.7 9.6 -2   .29  4.4  |

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

o20 = 1.77635683940025e-15

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

o21 = 1.77635683940025e-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 = | -10  21  -2   -4.2 6    |
      | 11   -22 2.8  3.4  -5.1 |
      | 7.9  -13 3.4  1.2  -7.6 |
      | -1.3 5.5 -1.1 -1.8 .89  |
      | -5.7 9.6 -2   .29  4.4  |

                 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 :