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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 = | .48+.26i .21+.95i  .58+.22i .41+.86i  .23+.016i .096+.44i .72+.11i 
      | .42+.12i .46+.6i   .34+.16i .23+.5i   .26+.97i  .68+.01i  .29+.74i 
      | .77+.11i .21+.065i .2+.97i  .64+.83i  .3+.12i   .84+.29i  .51+.7i  
      | .21+.84i .33+.15i  .76+.31i .83+.35i  .12+.22i  .96+.98i  .72+.92i 
      | .69+.84i .33+.18i  .41+.9i  .13+.75i  .51+.01i  .48+.48i  .94+.33i 
      | .98+.07i .9+.64i   .65+.97i .35+.91i  .55+.52i  .91+.24i  .18+.93i 
      | .86+.59i .73+.91i  .11+.94i .23+.081i .49+.43i  .79+.84i  .87+.01i 
      | .75+.92i .01+.58i  .6+.92i  .48+.63i  .7+.12i   .72+.71i  .24+.45i 
      | .78+.57i .83+.09i  .43+.34i .78+.44i  .41+.91i  .54+.18i  .012+.26i
      | .01+.76i .57+.52i  .31+.55i .74+.31i  .78+.7i   .8+.65i   .05+.87i 
      -----------------------------------------------------------------------
      .31+.4i  .29+.47i .93+.04i |
      .31+.22i .26+.74i .28+.18i |
      .88+.44i .14+.97i .44+.24i |
      .74+.67i .22+.59i .23+.56i |
      .59+.82i .78+.45i .58+.98i |
      .8+.92i  .46+.8i  .95+.27i |
      .85+.17i .47+.35i .74+.19i |
      .9+.26i  .31+.54i .62+.85i |
      .27+.8i  .91+.3i  .64+.1i  |
      .64+.67i .73+.47i .66+.91i |

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

o14 = | .18+.46i  .64+.03i |
      | .95+.52i  .96+.74i |
      | .27+.86i  .42+.19i |
      | .3+.34i   .16+.69i |
      | .85+.57i  .79+.68i |
      | .52+.61i  .29+i    |
      | .98+.13i  .4+.75i  |
      | .047+.19i .55+.11i |
      | .42+.26i  .65+.05i |
      | .94+.33i  .82+.34i |

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

o15 = | -.59+.25i  -.47+.053i |
      | -.47-1.9i  -.75+.39i  |
      | -1.2-.91i  -.56+.68i  |
      | .74-2.1i   -.82-1.9i  |
      | -.7-.15i   .47-.55i   |
      | 2.3+1.1i   1.1-.69i   |
      | .98-.85i   .17-.17i   |
      | -.31+.092i i          |
      | 1.4+3.7i   1.6+.61i   |
      | -1.7+.23i  .2+.56i    |

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

o16 = 1.21238184313208e-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 = | .12 1   .59  .82  .98 |
      | .59 .29 .72  .028 .21 |
      | .18 .74 .14  .092 .87 |
      | .82 .51 .025 .45  .8  |
      | .56 .42 .22  .45  .35 |

                 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 = | -.86 .28  .16  .28  1.2  |
      | -1.5 -1   2.9  -3.6 5.7  |
      | .93  1.4  -.96 .38  -1.9 |
      | 1.1  -.58 -1.6 .59  .045 |
      | 1.2  .65  -1.1 2.9  -4.8 |

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

o20 = 5.55111512312578e-16

o20 : RR (of precision 53)
i21 : norm(A'*A - I)

o21 = 8.88178419700125e-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 = | -.86 .28  .16  .28  1.2  |
      | -1.5 -1   2.9  -3.6 5.7  |
      | .93  1.4  -.96 .38  -1.9 |
      | 1.1  -.58 -1.6 .59  .045 |
      | 1.2  .65  -1.1 2.9  -4.8 |

                 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 :