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Macaulay2Doc :: solve

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 = | 2.2e-16  |
      | -2.2e-16 |
      | 0        |

                 3          1
o10 : Matrix RR    <--- RR
               53         53
i11 : norm oo

o11 = 2.22044604925031e-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 = | .99+.16i  .13+.9i  .01+.88i .13+.1i  .04+.81i .69+.59i  .18+.76i
      | .023+.23i .5+.66i  .31+.9i  .39+.46i .78+.92i .13+.45i  .36+.46i
      | .9+.45i   .82+.86i .36+.59i .09+.79i .21+.31i .62+.69i  .94+.69i
      | .51+.96i  .93+.81i .35+.38i .72+.63i .06+.73i .54+.04i  .65+.68i
      | .76+.31i  .4+.84i  .28+.47i .02+.84i .82+.75i .53+.96i  .06+.78i
      | .85+.49i  .48+.3i  .92+.22i .4+.62i  .85+.37i .99+.09i  .62+.04i
      | .34+.53i  .92+.1i  .89+.31i .35+.69i .71+.03i .73+.59i  .46+.17i
      | .26+.81i  .36+.32i .41+.15i .31+.27i .67+.26i .039+.4i  .68+.48i
      | .81+.34i  .06+.72i .1+.41i  .38+.48i .02+i    .21+.9i   .35+.19i
      | .24+.18i  .95+.16i .86+.25i .17+.57i .4+.58i  .078+.31i .85+.64i
      -----------------------------------------------------------------------
      .6+.19i  .66+.26i  .083+.38i |
      .81+.96i .41+.59i  .46+.63i  |
      .97+.02i .71+.9i   .76+.42i  |
      .77+.44i .02+.67i  .3+.042i  |
      .32+.27i .35+.39i  .98+.77i  |
      .16+.71i .54+.26i  .09+.54i  |
      .15+.65i .95+.58i  .05+.52i  |
      .35+.73i .97+.44i  .96+.18i  |
      .67+.01i .25+.25i  .53+.45i  |
      .48+.21i .28+.002i .028+.3i  |

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

o14 = | .32+.64i  .77+.93i |
      | .28+.86i  .66+.78i |
      | .21+.003i .53+.61i |
      | .44+.52i  .77+.44i |
      | .98+.36i  .42+.62i |
      | .92+.63i  .37+.54i |
      | .78+.71i  .99+.92i |
      | .48+.084i .93+.81i |
      | .23+.36i  .29+.18i |
      | .62+.95i  .41+.99i |

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

o15 = | -.07-.69i  -.73-1.4i  |
      | .22-.66i   1.5-1.8i   |
      | -1+.12i    -3.1-1.1i  |
      | 1.2-.21i   2.4+.43i   |
      | 1.2+.82i   1.1+2i     |
      | -.23+.083i -.44-.091i |
      | .45-.52i   1.7+.14i   |
      | -.35+.82i  -.99+1.1i  |
      | .16+1.1i   .67+2.7i   |
      | -.7-1.4i   -.82-2.8i  |

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

o16 = 1.35064460289285e-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 = | .058 .94  .83 .35  .87 |
      | .49  .013 .72 .068 .47 |
      | .44  .7   .18 .72  .11 |
      | .35  .73  .55 .71  .56 |
      | .72  .69  .25 .84  .43 |

                 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 = | .69  1    1    -3.5 1.7  |
      | 2    -.14 2.3  -3.6 .24  |
      | -.57 1.3  1.9  1.6  -2.8 |
      | -2.2 -.7  -1.4 5.1  -.95 |
      | .41  -.91 -3.8 .62  2.6  |

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

o20 = 5.96744875736022e-16

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

o21 = 1.22124532708767e-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 = | .69  1    1    -3.5 1.7  |
      | 2    -.14 2.3  -3.6 .24  |
      | -.57 1.3  1.9  1.6  -2.8 |
      | -2.2 -.7  -1.4 5.1  -.95 |
      | .41  -.91 -3.8 .62  2.6  |

                 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 :