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 = | 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 = | .54+.16i .51+.72i  .89+.22i .42+.76i  .81+.28i  .26+.48i  .81+.38i
      | .62+.67i .83+.04i  .59+.49i .16+.46i  .85+.19i  .51+.76i  .41+.41i
      | .73+.77i .018+.41i .61+.97i .58+.66i  .093+.15i .01+.13i  .62+.45i
      | .41+.82i .66+.97i  .17+.63i .029+.26i .009+.46i .06+.85i  .23+.6i 
      | .42+.86i .59+.68i  .1+i     .12+.8i   .45+.37i  .16+.082i .76+.66i
      | .74+.77i .24+.047i .75+.8i  .15+.78i  .06+.96i  .94+.27i  .84+.64i
      | .94+.99i .79+.18i  .42+.11i .11+.12i  .26+.008i .12+.19i  .81+.73i
      | .35+.71i .99+.43i  .13+.85i .45+.96i  .47+.4i   .12+.53i  .21+.85i
      | .58+.64i .05+.66i  .07+.64i .67+.19i  .59+.26i  .14+.041i .54     
      | .29+.76i .19+.47i  .71+.79i .35+.57i  .11+.13i  .11+.21i  .12+.86i
      -----------------------------------------------------------------------
      .24+.22i .27+.28i .1+.6i    |
      .16+.26i .28+.34i .34+i     |
      .81      .41+.4i  .66+.24i  |
      .89+.23i .43+.72i .51+.13i  |
      .74+.87i .56+.05i .28+.87i  |
      .87+.14i .62+.86i .33+.13i  |
      .78+.46i .9+.13i  .55+.48i  |
      .66+.64i .85+.29i .24+.69i  |
      .01+.27i .2+.093i .22+.61i  |
      .39+.69i .17+.81i .095+.47i |

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

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

o14 = | .052+.35i .9+.83i   |
      | .15+.34i  .53+.73i  |
      | .46+.082i .5+.42i   |
      | .2+.57i   .13+.31i  |
      | .65+.91i  .19+.34i  |
      | .08+.94i  .023+.49i |
      | .88+.95i  .52+.44i  |
      | .92+.71i  .6+.93i   |
      | .19+.5i   .77+.63i  |
      | .45+.75i  .67+.34i  |

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

i15 : x = solve(A,b)

o15 = | 1.4+.04i   .083+.19i |
      | -.12-.003i -.62+.9i  |
      | -.36+.26i  -.41-.72i |
      | .43-.19i   .6+.3i    |
      | .04+.63i   1.6+1.4i  |
      | .2+.16i    -.82-1.5i |
      | -1+.01i    1.3+.51i  |
      | 1.6-.08i   -.14-1.4i |
      | .019+.31i  .78-1.1i  |
      | -1.4-.46i  -.98+1.6i |

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

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

o16 = 1.08246744900953e-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 = | .028 .46  .55 .59 .93 |
      | .89  .8   .61 .67 .44 |
      | .18  .52  .91 .61 .9  |
      | .29  .84  1   .15 .99 |
      | .27  .017 .97 .85 .15 |

                 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 = | -21 1.6  36   -12 -11  |
      | 25  -.39 -44  15  12   |
      | .26 -.58 -2.1 1.7 1.5  |
      | 8.9 .083 -14  3.6 4.3  |
      | -16 .43  31   -10 -9.2 |

                 5          5
o19 : Matrix RR    <--- RR
               53         53

i20 : norm(A*A' - I)

o20 = 8.88178419700125e-15

o20 : RR (of precision 53)

i21 : norm(A'*A - I)

o21 = 1.06581410364015e-14

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 = | -21 1.6  36   -12 -11  |
      | 25  -.39 -44  15  12   |
      | .26 -.58 -2.1 1.7 1.5  |
      | 8.9 .083 -14  3.6 4.3  |
      | -16 .43  31   -10 -9.2 |

                 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

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` :