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 = | .51+.47i .1+.58i  .46+.2i   .26+.96i  .17+.001i .7+.67i    .75+.97i 
      | .3+.22i  .63+.49i .2+.35i   .99+.68i  .61+.09i  .4+.49i    .55+.48i 
      | .54+.64i .39+.6i  .69+.06i  .91+.04i  .84+.26i  .69+.79i   .068+.34i
      | .55+.93i .34+.19i .44+.021i .43+.34i  .89+.67i  .8+.72i    .9+.04i  
      | .68+.48i .13+.36i .39+.8i   .033+.22i .5+.67i   .064+.038i .92+.16i 
      | .79+.89i .54+.43i .85+.88i  .022+.19i .35+.39i  .54+.72i   .69+.58i 
      | .8+.25i  .21+.48i .66+.56i  .52+.87i  .22+.98i  .24+.82i   .73+.32i 
      | .66+.54i .54+.94i .67+.83i  .18+.15i  .43+.62i  .12+.084i  .83+.98i 
      | .74+.81i .93+.88i .39+.82i  .82+.54i  .92+.01i  1+.54i     .72+.81i 
      | .82+.6i  .74+.16i .87+.77i  .32+.66i  .54+.6i   .5+.2i     1+.55i   
      -----------------------------------------------------------------------
      .71+.55i .73+.31i  .9+.41i   |
      .32+.12i .58+.43i  .98+.77i  |
      .35+.94i .71+.68i  .63+.08i  |
      .57+.92i .03+.74i  .49+.71i  |
      .4+.63i  .046+.11i .02+.65i  |
      .96+.06i .82+.18i  .94+.91i  |
      1+.18i   .52+.07i  .47+.51i  |
      .94+.73i .91       .12+.076i |
      .87+.42i .7+.21i   .18+.95i  |
      .15+.72i .73+.68i  .72+.3i   |

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

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

o14 = | .83+.43i   .02+.56i  |
      | .04+.7i    .99+.43i  |
      | .81+.38i   .11+.56i  |
      | .05+.55i   .052+.34i |
      | .25+.41i   .49+.27i  |
      | .49+.011i  .99+.93i  |
      | .24+.98i   .14+.76i  |
      | .97+.94i   .84+.67i  |
      | .096+.096i .32+.92i  |
      | .82+.72i   .7+.05i   |

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

i15 : x = solve(A,b)

o15 = | .73+.67i  .06+.21i  |
      | -2-.59i   .77-.76i  |
      | .83+.07i  -.51+.47i |
      | .33+1.6i  -.6+.48i  |
      | .2+1.3i   .28-.76i  |
      | -.79-1.6i -.26+.36i |
      | 2.9-.87i  .21+.22i  |
      | -.74+2i   -.77+.04i |
      | 1.1-2.9i  .88+.64i  |
      | -1.7+.83i .87-.69i  |

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

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

o16 = 1.48952049194836e-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 = | .25 .57  .47 .069 .021 |
      | .81 .054 .11 .29  .97  |
      | .42 .39  .64 .68  .84  |
      | .67 .67  .52 .56  .33  |
      | .21 .78  .99 .83  1    |

                 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 = | 1.1  .08  2.4  .13  -2.2 |
      | -1.5 1.3  -7.1 2.2  4    |
      | 3.7  -1.6 7.4  -3   -3.8 |
      | -2.8 -1.1 .26  2.1  .16  |
      | -.46 1.4  -2.5 -.52 2    |

                 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 = 1.76247905159244e-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 = | 1.1  .08  2.4  .13  -2.2 |
      | -1.5 1.3  -7.1 2.2  4    |
      | 3.7  -1.6 7.4  -3   -3.8 |
      | -2.8 -1.1 .26  2.1  .16  |
      | -.46 1.4  -2.5 -.52 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` :