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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 = | 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 = | .9+.08i  .56+.83i .51+.54i   .099+.002i .96+.35i  .39+.35i  .12+.72i
      | .4+.74i  .22+.86i .06+.6i    .72+.87i   .35+.029i .2+.72i   .53+.61i
      | .89+.51i .17+.68i .64+.84i   .072+.5i   .88+.5i   .97+.77i  .68+.62i
      | .21+.42i .66+.54i .055+.44i  .89+.34i   .76+.8i   .33+.43i  .71+.74i
      | .07+.43i .42+.84i .98+.8i    .16+.047i  .37+.31i  .065+.26i .35+.94i
      | .72+.5i  .43+.88i .75+.81i   .95+.68i   .33+.58i  .19+.83i  .14+.84i
      | .42+.97i .95+.6i  .1+.74i    .68+.54i   .07+.58i  .05+.96i  .73+.32i
      | .28+.71i .69+.4i  .44+.71i   .29+.44i   .4+.33i   .75+.7i   .91+.69i
      | .28+.31i .31+.53i .97+.69i   .59+.93i   .98+.07i  .73+.73i  .66+.13i
      | .17+.31i .59+.82i .032+.018i .84+.96i   .48+.11i  .75+.59i  1+.94i  
      -----------------------------------------------------------------------
      .86+.38i .87+.15i  .28+.74i  |
      .13+.38i .092+.18i .86+.8i   |
      .97+.75i .96+.67i  .83+.58i  |
      .43+.61i .84+.82i  .42+.01i  |
      .79+.71i .08+.77i  .3+.57i   |
      .64+.88i .96+.14i  .33+.025i |
      .65+.4i  .82+.96i  .53+.71i  |
      .03+.85i .46+.14i  .047+.15i |
      .56+.54i .93+.32i  .7+.85i   |
      .32+.72i .36+.29i  .82+.93i  |

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

o14 = | .35+.25i .78+.12i  |
      | .53+.52i .81+.87i  |
      | .33+.95i .6+.2i    |
      | .64+.81i .59+.86i  |
      | .67+.48i .3+.84i   |
      | .01+.83i .38+.39i  |
      | .85+.93i .86+.73i  |
      | .23+.15i .35+.84i  |
      | .85+.56i .53+.24i  |
      | .55+.32i .44+.041i |

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

o15 = | .15-.12i  1.6-.3i   |
      | -.72+.28i -.09+.7i  |
      | .49-.79i  .15+.019i |
      | 1.3+.45i  .73+.04i  |
      | -.64+.76i .41+1.2i  |
      | .94+.74i  -.2+.16i  |
      | .37-1.1i  .1-.59i   |
      | -.71+1.9i -.65+.46i |
      | .83-1.2i  .01-1.2i  |
      | -.83-.9i  -.6-.33i  |

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

o16 = 7.30135198854559e-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 = | .072 .39 .64  .97 .85 |
      | .99  .88 .019 .23 .23 |
      | .23  .39 .041 .45 .73 |
      | .77  .98 .93  .42 .68 |
      | .96  .39 .77  .66 1   |

                 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 = | -.26 .71  -.72 -.66 1     |
      | .077 .32  .65  .96  -1.3  |
      | .091 -.68 -1   .89  .22   |
      | 2    1.2  -1.4 -1.3 -.039 |
      | -1.2 -1.1 2.2  .44  .36   |

                 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 = 4.9960036108132e-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 = | -.26 .71  -.72 -.66 1     |
      | .077 .32  .65  .96  -1.3  |
      | .091 -.68 -1   .89  .22   |
      | 2    1.2  -1.4 -1.3 -.039 |
      | -1.2 -1.1 2.2  .44  .36   |

                 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 :