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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 = | .16+.31i .5+.82i  .97+.89i  .55+.75i  .5+.022i .95+.54i .8+.54i 
      | .67+.91i .57+.07i .9+.61i   .25+.12i  .41+.45i .65+.02i .71+.13i
      | .26+.78i .43+.84i .13+.055i .25+.81i  .54+.02i .9+.29i  .86+.05i
      | .39+.89i .13+.02i .97+.45i  .61+.37i  .62+.46i .07+.32i .85+.06i
      | .76+.15i .07+.8i  .1+.2i    .49+.77i  .96+.64i .29+.11i .97+.83i
      | .11+.68i .69+.67i .74+.49i  .084+.15i .16+.87i .74+.93i .41+.3i 
      | .93+.86i .38+.6i  .58+.51i  .79+.9i   .51+.46i .95+i    .06+.74i
      | .63+.03i .67+.84i .46+.53i  .42+.031i .5+.87i  .7+.56i  .37+.14i
      | .07+.73i .7+.47i  .092+.35i 1+.35i    .18+.79i .7+.11i  .9+.2i  
      | .81+.35i .37+.83i .29+.83i  .19+.028i .84+.71i .27+.4i  .33+.29i
      -----------------------------------------------------------------------
      .69+.64i .34+.38i  .57+.1i   |
      .67+.96i .34+.27i  .99+.2i   |
      .39+.99i .23+.64i  .72+.15i  |
      .29+.8i  .095+.12i .63+.88i  |
      .6+.51i  .49+.77i  .26+.92i  |
      .37+.74i .14+.35i  .31+.47i  |
      .93+.47i .15+.22i  .41+.059i |
      .24+.26i .96+.26i  .66+.82i  |
      .03+.77i .23+.38i  .77+.63i  |
      .4+.95i  .05+.53i  .61+.46i  |

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

o14 = | .93+.16i .48+.16i   |
      | .8i      .75+.55i   |
      | .72+.03i .44+.17i   |
      | .95+.67i .32+.43i   |
      | .77+.31i .31+.32i   |
      | .62+.13i .079+.019i |
      | .17+.4i  .85+.3i    |
      | .7+.02i  .83+.97i   |
      | .63+.25i .77+.86i   |
      | .33+.81i .93+.3i    |

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

o15 = | -.01+1.8i .69-.12i  |
      | -.52+1.1i .53-.17i  |
      | .6+.48i   .55+.26i  |
      | .99-.58i  .65-.15i  |
      | .23-.44i  .5-.31i   |
      | .36-1.1i  -.85-.08i |
      | .34+.97i  -.25-.68i |
      | -.63-2i   -1.7-.25i |
      | .67-.63i  .76-.33i  |
      | -.95+.41i .54+1.6i  |

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

o16 = 1.33688555545767e-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 = | .94 .23 .92 .65 .31 |
      | .28 .4  .69 .65 .45 |
      | .78 .67 .46 .73 .76 |
      | .89 .3  .24 .49 .62 |
      | .5  .45 .81 .22 .64 |

                 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  -2   .9   -.17 -.024 |
      | .92  -3.3 4.9  -4.3 .27   |
      | .53  .43  -.77 -.5  .84   |
      | -.11 2.2  -.5  .8   -1.7  |
      | -2.1 2.6  -3   3.5  .89   |

                 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 = 7.7715611723761e-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 = | 1.1  -2   .9   -.17 -.024 |
      | .92  -3.3 4.9  -4.3 .27   |
      | .53  .43  -.77 -.5  .84   |
      | -.11 2.2  -.5  .8   -1.7  |
      | -2.1 2.6  -3   3.5  .89   |

                 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 :