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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 = | .8+.35i  .95         .22+.3i   .58+.51i  .27+.001i .44+i    
      | .78+.77i .34+.11i    .38+.57i  .09+.66i  .08+.64i  .68+.18i 
      | .94+.03i .72+.61i    .86+.57i  .13+.002i .1+.61i   .25+.31i 
      | .26+.44i .0023+.041i .92+.41i  .27+.55i  1+.28i    .27+.067i
      | .67+.83i .72+.64i    .43+.067i .12+.83i  .39+.56i  .73+.26i 
      | .31+.63i .04+.74i    .9+.54i   .59+.52i  .81+.28i  .88+.86i 
      | 1+.44i   .71+.25i    .33+.55i  .62+.52i  .61+.71i  .99+.81i 
      | .75+.32i .11+.24i    .39+.063i .73+.02i  .86+.4i   .58+.22i 
      | .81+.59i .95+.58i    .66+.43i  .16+.21i  .62+.8i   .29+.65i 
      | .55+.52i .43+.53i    .84+.12i  .32+.26i  .59+.13i  .38+.81i 
      -----------------------------------------------------------------------
      .19+.019i .67+.42i  .6+.68i   .02+.53i |
      .28+.071i .33+.027i .79+.21i  .4+.16i  |
      .66+.52i  .1+.55i   .18+.21i  .66+.77i |
      .58+.91i  .12+.33i  .71+.56i  .55+.93i |
      .25+.91i  .14+.35i  .15+.96i  .67+.06i |
      .57+.03i  .73+.07i  .85+.32i  .29+.95i |
      .1+.22i   .18+.23i  .51+.7i   .68+.85i |
      .67+.11i  .14+.24i  .7+.63i   .29+i    |
      .96+.54i  .5+.13i   .46+.085i .7+.11i  |
      .34+.48i  .66+.32i  .52+.27i  .098+.3i |

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

o14 = | .16+.34i .71+.55i |
      | .5+.72i  .11+.44i |
      | .94+.14i .78+.58i |
      | .63+.82i .56+.67i |
      | .8+.83i  .85+.1i  |
      | .54+.37i .3+.47i  |
      | .38+.28i .68+.08i |
      | .66+.2i  .66+.82i |
      | .88+.89i .49+.3i  |
      | .61+.3i  .59+.35i |

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

o15 = | 1.9+.87i  2.3+.83i  |
      | -1.1-.69i -1.2-.42i |
      | 1.4-.67i  .88-.4i   |
      | -.37+.35i -.5+.41i  |
      | -.38+.65i -.19+.76i |
      | -1.5-.43i -2.1-.15i |
      | .28+.069i -.53+.08i |
      | -.06+3.2i 1.6+3.3i  |
      | .13-2.7i  -.59-3.1i |
      | .44+.78i  1.2+.07i  |

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

o16 = 1.49365231817119e-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 = | .82   .38 .16 .87 .35 |
      | .49   .65 .51 .68 .15 |
      | .19   .54 .63 .2  .29 |
      | .33   .22 .18 .72 .14 |
      | .0094 .63 .7  .94 .42 |

                 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 = | .013 -.56 2.1  2.7  -2.1 |
      | 2.4  4.5  -5   -9.9 3.2  |
      | -3.3 -2.6 5.5  9.1  -3.2 |
      | -.54 .022 -.78 1.8  .37  |
      | 3.1  -2.4 .036 -4.1 2.1  |

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

o20 = 6.66133814775094e-16

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 = | .013 -.56 2.1  2.7  -2.1 |
      | 2.4  4.5  -5   -9.9 3.2  |
      | -3.3 -2.6 5.5  9.1  -3.2 |
      | -.54 .022 -.78 1.8  .37  |
      | 3.1  -2.4 .036 -4.1 2.1  |

                 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 :