solve -- solve a linear equation

Synopsis

Usage:
x = solve(A,b)
Inputs:
- A, a matrix, or a mutable matrix of size m by n over either a finite field ZZ/p, RR or CC
- b, the same type of matrix, over the same ring, of size m by r
Optional inputs:
- MaximalRank => an integer,
- ClosestFit => a Boolean value, default value false, whether to use the least squares method
- MaximalRank => a Boolean value, default value false, whether to assume the matrix has maximal rank, in case the least squares method is used
- Precision => ... (missing documentation),
Outputs:
- x, the same type of matrix, over the same ring, of size n by r, such that Ax=b

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 = | .61+.78i .13+.94i  .24+.98i  .63+.42i .23+.87i  .16+.92i .59+.08i
      | .57+.73i .53+.96i  .81+.83i  .57+.29i .48+.041i .33+.33i .61+.59i
      | .83+.39i .59+.93i  .55+.02i  .6+.6i   .46+.47i  .94+.12i .94+.67i
      | .31+.94i .68+.63i  .66+.21i  .19+.41i .36+.76i  .05+.86i .9+.85i 
      | .62+.05i .01+.86i  .4+.25i   .74+.62i .63+.18i  .99+.96i .87+.29i
      | .56+.94i .78+.22i  .073+.14i .76+.27i .16+.55i  .58+.73i .63+.33i
      | .26+.27i .037+.37i .53+.01i  .98+.83i .97+.64i  .92+.54i .99+.45i
      | .46+.6i  .04+.56i  .45+.37i  .89+.5i  .13+.26i  .55+.28i .86+.81i
      | .44+.8i  .24+.76i  .65+.22i  .21+.8i  .089+.45i .38+.75i .41+.1i 
      | .32+.8i  .62+.34i  .34+.021i .74+.13i .57+.94i  .58+.69i .05+.98i
      -----------------------------------------------------------------------
      .46+.29i .97+.24i  .037+.5i   |
      .93+.2i  .49+.55i  .053+.055i |
      .05+.96i .69+.29i  .08+.58i   |
      .6+.61i  .25+.66i  .45+.98i   |
      .12+.64i .31+.38i  .68+.81i   |
      .34+.49i .006+.27i .42+.96i   |
      .12+.14i .23+.31i  .97+.98i   |
      .9+.35i  .31+.04i  .24+.14i   |
      .58+.9i  .41+.026i .64+.05i   |
      .27+.75i .33+.73i  .31+.62i   |

                 10          10
o13 : Matrix CC     <--- CC
               53          53

i14 : b = random(CC^n,CC^2)

o14 = | .64+.38i  .84+.36i |
      | .42+.31i  .47+.78i |
      | .041+.41i .34+.36i |
      | .78+.53i  .94+.97i |
      | .59+.94i  .41+.84i |
      | .41+.23i  .35+i    |
      | .66+.83i  .17+.59i |
      | .76+.83i  .84+.32i |
      | .59+.82i  .48+.36i |
      | .79+.64i  .57+.81i |

                 10          2
o14 : Matrix CC     <--- CC
               53          53

i15 : x = solve(A,b)

o15 = | -.005-.3i  .75+1.3i  |
      | -.67-1.3i  1.6-.74i  |
      | -.3+1.2i   -1.1-.56i |
      | .03+.77i   .15+.095i |
      | -.2-.17i   -1.2-.51i |
      | .094-.044i -1.1-.64i |
      | -.58-.17i  .36-2i    |
      | 1.1+.63i   -.75+.9i  |
      | .47+.11i   .54+.53i  |
      | .71-.61i   2.3+1.3i  |

                 10          2
o15 : Matrix CC     <--- CC
               53          53

i16 : norm ( matrix A * matrix x - matrix b )

o16 = 1.80731214395321e-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 = | .69 1    .54 .15  .12 |
      | .3  .82  .44 .079 .83 |
      | .71 .25  .35 .27  .29 |
      | .56 .26  .2  .33  .31 |
      | .33 .072 .84 .19  .44 |

                 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 = | -.38 .11  4.5  -2.9 -1   |
      | 1    .21  -1.9 1.2  -.25 |
      | .55  -.45 -.68 -.42 1.5  |
      | .68  -1.3 -6.9 7.9  1.3  |
      | -1.2 1.3  1.2  -.62 -.29 |

                 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 = 9.99200722162641e-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 = | -.38 .11  4.5  -2.9 -1   |
      | 1    .21  -1.9 1.2  -.25 |
      | .55  -.45 -.68 -.42 1.5  |
      | .68  -1.3 -6.9 7.9  1.3  |
      | -1.2 1.3  1.2  -.62 -.29 |

                 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

Ways to use `solve` :

solve(Matrix,Matrix)
solve(MutableMatrix,MutableMatrix)

solve -- solve a linear equation

Synopsis

Description

Caveat

See also

Ways to use solve :

Ways to use `solve` :