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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 |
      | 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 = | .55+.16i .85+.16i .08+.8i    .63+.18i  .93+.74i  .86+.87i  .04+.43i
      | .13+.71i .4+.77i  .25+.98i   .79+.74i  .3+.71i   .7+.99i   .77+.38i
      | .25+.33i .92+.65i .047+.026i .15+.26i  .38+.19i  .6+.91i   .33+.16i
      | .88+.59i .68+.87i .43+.76i   .35+.91i  .14+.87i  .1+.52i   .14+.1i 
      | .29+.7i  .97+.95i .6+.22i    .23+.44i  .04+.57i  .54+.46i  .1+.46i 
      | .1+.6i   .66+.73i .72+.26i   .78+.63i  .43+.36i  .3+.014i  .84+.22i
      | .32+.61i .74+.57i .82+.51i   .23+.36i  .12+.47i  .77+.19i  .86+.75i
      | .8+.95i  .31+.66i .44+.098i  .09+.91i  .032+.21i .019+.18i .94+.58i
      | .55+.73i .26+.91i .26+.75i   .28+.36i  .92+.93i  .8+.32i   .42+.46i
      | .41+.45i .62+.15i .77+.75i   .077+.24i .26+.76i  .77+.69i  .19+.3i 
      -----------------------------------------------------------------------
      .71+.8i  .02+.34i .91+.72i  |
      .53+.4i  .46+.33i .09+.55i  |
      .58+.16i .75+.14i .03+.56i  |
      .8+.89i  .36+.13i .15+.51i  |
      .48+.34i .61+.55i .62+.09i  |
      .84+.99i .92+.7i  .25+.69i  |
      .76+.54i .63+.98i .12+.57i  |
      .4+.15i  .2+.98i  .78+.51i  |
      .03+.94i .63+.73i .47+.088i |
      .93+.89i .4+.043i .065+.21i |

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

o14 = | .34+.63i .6+.8i   |
      | .58+.97i .49+.54i |
      | .64+.24i .73+.04i |
      | .18+.97i .7+.37i  |
      | .92+.98i .2+.41i  |
      | .83+.89i .19+.87i |
      | .62+.06i .78+.51i |
      | .71+.93i .91+.82i |
      | .07+.69i .06+.62i |
      | .56+.78i .1+.74i  |

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

o15 = | .26-.61i   -.21-.29i |
      | .05+.0067i 1.4-.33i  |
      | .15+1.2i   -1.5-.51i |
      | 1-.11i     -.88+.13i |
      | -.45+.58i  .45-.59i  |
      | .74-.52i   .005+.45i |
      | -.016-.17i 1+2i      |
      | .29-.52i   1.8+.76i  |
      | -.49-.097i -1.5-1.7i |
      | -.18+.67i  -.57+.79i |

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

o16 = 6.66133814775094e-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 = | .28  .86 .039 .077 .58   |
      | .077 .41 .55  .82  .0023 |
      | .055 .4  .046 .98  .65   |
      | .33  .21 .076 .7   .83   |
      | .17  .59 .41  .78  .63   |

                 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 = | 1.6   3    -1.6 3.6  -4.6 |
      | 1.3   .44  .71  -.99 -.6  |
      | -1.2  -.63 -2   -.41 3.7  |
      | .0074 1.1  1.1  .42  -1.7 |
      | -.85  -2.2 -.33 -.31 3.1  |

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

o20 = 2.77555756156289e-16

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

o21 = 4.44089209850063e-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 = | 1.6   3    -1.6 3.6  -4.6 |
      | 1.3   .44  .71  -.99 -.6  |
      | -1.2  -.63 -2   -.41 3.7  |
      | .0074 1.1  1.1  .42  -1.7 |
      | -.85  -2.2 -.33 -.31 3.1  |

                 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 :