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
Outputs:
- x, the same type of matrix, over the same ring, of size n by r, such that Ax=b
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

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 = | -1.1e-16 |
      | -7.8e-16 |
      | 0        |

                 3          1
o10 : Matrix RR    <--- RR
               53         53

i11 : norm oo

o11 = 7.7715611723761e-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 = | .78+.12i .96+.4i  .42+.79i  .72+.87i  .26+.32i .87+.26i  .3+.48i 
      | .72+.48i .49+.37i .34+.002i .89+.42i  .56+.69i .093+.38i .42+.68i
      | .1+.63i  .2+.44i  .93+.74i  .055+.43i .86+.12i .79+.02i  .22+.28i
      | .26+.61i .41+.44i .55+.92i  .45+.89i  .5+.23i  .32+.1i   .22+.75i
      | .43+.34i .56+.17i .87+.22i  .024+.48i .85+.85i .68+.27i  .8+.44i 
      | .64+.83i .17+.91i .056+.21i .25+.45i  .22+.91i .57+.1i   .33+.59i
      | .68+.3i  .96+.93i .45+.97i  .45+.8i   .9+.89i  .8+.91i   .11+i   
      | .14+.38i .1+.85i  .65+.62i  .06+.76i  .59+.27i .75+.93i  .13+.82i
      | .74+.07i .91+.76i .38+.8i   .13+.41i  .54+.76i .58+.33i  .77+.22i
      | .59+.73i .3+.87i  .3+.83i   .89+.15i  .18+.37i .34+.078i .72+.78i
      -----------------------------------------------------------------------
      .72+.49i  .53+.37i .27+.68i  |
      .25+.94i  .75+.7i  .03+.57i  |
      .81+.75i  .92+.1i  .82+.81i  |
      .32+.95i  .67+.39i .22+.16i  |
      .85+.45i  .06+.61i .66+.99i  |
      .85+.76i  .52+.07i .65+.77i  |
      .1+.52i   .92+.67i .74+.02i  |
      .66+.34i  .65+.12i .82+.6i   |
      .79+.56i  .62+.55i .63+.11i  |
      .35+.023i .38+.75i .068+.24i |

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

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

o14 = | .69+.22i .14+.89i   |
      | .57+.82i .75+.23i   |
      | .41+.9i  .52+.51i   |
      | .82+.77i .42+.37i   |
      | .43+.34i .12+.2i    |
      | .73i     .25+.44i   |
      | .31+i    .049+.024i |
      | .95+.19i .84+.59i   |
      | .63+.41i .38+.46i   |
      | .95+.27i .63+.17i   |

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

i15 : x = solve(A,b)

o15 = | -.71+.78i -3.3+1.5i |
      | .87-.59i  2-2.6i    |
      | -.19+.32i -.17+1.1i |
      | -.7-.3i   .71-.25i  |
      | -.04+1.1i .12+.5i   |
      | -1.1+.49i -1.9+1.1i |
      | .69-1.7i  .52-.9i   |
      | .88+.52i  .58+.27i  |
      | .92-.44i  .8-.15i   |
      | -.38-.51i .2-1.2i   |

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

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

o16 = 1.05325004057301e-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 = | .34 .091 .56  .6  .22 |
      | .85 .23  .55  .54 .63 |
      | .47 .1   .39  .74 .75 |
      | .7  .91  .73  .75 .66 |
      | .44 .004 .098 .32 .13 |

                 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 = | -.59 1.2  -1   -.14 1.7 |
      | -.83 -.96 -.31 1.5  .49 |
      | 2    1.8  -1.2 -.53 -3  |
      | .7   -2.6 1.1  .45  2.5 |
      | -1.3 .94  1.5  -.27 -2  |

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

i20 : norm(A*A' - I)

o20 = 4.44089209850063e-16

o20 : RR (of precision 53)

i21 : norm(A'*A - I)

o21 = 4.44089209850063e-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 = | -.59 1.2  -1   -.14 1.7 |
      | -.83 -.96 -.31 1.5  .49 |
      | 2    1.8  -1.2 -.53 -3  |
      | .7   -2.6 1.1  .45  2.5 |
      | -1.3 .94  1.5  -.27 -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` :