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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 = | .17+.55i .69+.13i .64+.1i   .98+.59i  .35+.7i   .13+.83i .018+.28i
      | .16+.39i .28+.66i .31+.53i  .95+.73i  .16+.77i  .23+.45i .56+.38i 
      | .28+.27i .77+.96i .7+.35i   .81+.53i  .66+.22i  .21+.92i .02+.71i 
      | .08+.98i .6+.59i  .29+.85i  .55+.87i  .068+.31i .6+.34i  .67+.67i 
      | .4+.18i  .16+.52i .082+.38i .75+.08i  .1+.18i   .81+.93i .32+.88i 
      | .43+.35i .45+.55i .4+.64i   .89+.65i  .22+.1i   .14+.26i .013+.45i
      | .71+.09i .49+.6i  .61+.81i  .33+.19i  .54+.67i  .93+.64i .89+.28i 
      | .75+.02i .18+.93i .83+.22i  .46+.039i .52+.26i  .39+.13i .045+.44i
      | .15+.51i .74+.37i .34+.25i  .25+.57i  .5+.49i   .23+.35i .3+.71i  
      | .39+.75i .27+.59i .019+.3i  .55+.32i  .92+.05i  1+.29i   .24+.21i 
      -----------------------------------------------------------------------
      .59+.76i  .11+.47i .31+.87i  |
      .03+.9i   .62+.42i .5+.84i   |
      .19+.7i   .62+.1i  .32+.87i  |
      .56+.12i  .63+.72i .98+.56i  |
      .27+.052i .88+.7i  .91+.79i  |
      .03+.94i  .6+.31i  .15+.72i  |
      .32+.48i  .56+.53i .03+.98i  |
      .87+.12i  .38+.4i  .35+.006i |
      .11+.21i  .39+.57i .57+.34i  |
      .42+.76i  .53+.38i .86+.99i  |

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

o14 = | .56+.33i  .25+.07i  |
      | .78+.64i  .42+.35i  |
      | .24+.045i .4+.95i   |
      | .96+.48i  .072+.27i |
      | .8+.46i   .74+.6i   |
      | .53+.33i  .69+.16i  |
      | .87+.58i  .52+.31i  |
      | .62+.47i  .85+.13i  |
      | .25+.16i  .95+.58i  |
      | .74+.43i  .49+.52i  |

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

o15 = | .025-.081i  .011+.23i |
      | .49-.36i    .59-.36i  |
      | -.48+.27i   1+.77i    |
      | -.073+.082i -.55+1.1i |
      | -.59-.31i   -.33+i    |
      | .15+.12i    .045-.31i |
      | -.08+.79i   .44+.21i  |
      | .7-.11i     -1.1-1.9i |
      | 1.2-.53i    -.06-2.6i |
      | -.25-.072i  .64+1.2i  |

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

o16 = 7.85046229341888e-16

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   .58 .6  .099  .74 |
      | .85   .7  .74 .13   .47 |
      | .11   .68 .08 .37   .53 |
      | .12   .26 .4  .86   .31 |
      | .0045 .53 .74 .0091 .65 |

                 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   .26  -.27 -.0094 -.93 |
      | -1.9  2    1.5  -.73   -.2  |
      | -.7   .87  -1.2 .47    .93  |
      | -.042 -.12 .096 1.2    -.49 |
      | 2.3   -2.6 .15  .049   .66  |

                 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 = 6.66133814775094e-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   .26  -.27 -.0094 -.93 |
      | -1.9  2    1.5  -.73   -.2  |
      | -.7   .87  -1.2 .47    .93  |
      | -.042 -.12 .096 1.2    -.49 |
      | 2.3   -2.6 .15  .049   .66  |

                 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 :