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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 = | .38+.47i .6+.54i   .35+.72i  .94+.1i   .51+.76i .88+.25i  .37+.02i 
      | .62+.91i .69+.34i  .61+.08i  .16+.006i .17+.9i  .83+.99i  .12+.28i 
      | .66+.11i .25+.12i  .66+.91i  .1+.64i   .21+.12i .96+.39i  .25+.38i 
      | .81+.09i .69+.69i  .87+.38i  .32+.71i  .23+.41i .78+.64i  .69+i    
      | .86+.53i .83+.15i  .37+.22i  .59+.43i  .97+i    .43+.22i  .19+.97i 
      | .27+.64i .096+.13i .66+.54i  .28+.2i   .59+.7i  .89+.83i  .67+.41i 
      | .94+.04i .94+.51i  .31+.058i .55+.14i  .34+.3i  .55+.86i  .6+.14i  
      | .64+.34i .19+.19i  .59+.75i  .84+.03i  .77+.98i .42+.038i .95+.02i 
      | .26+.14i .24+.72i  .67+.44i  .33+.4i   .87+.67i .82+.92i  .32+.099i
      | .99+.11i .64+.82i  .26+.42i  .46+.63i  .25+.42i .5+.24i   .51+.93i 
      -----------------------------------------------------------------------
      .49+.23i .016+.46i .58+.27i |
      .8+i     .41+.47i  .96+.78i |
      .71+.03i .15+.27i  .86+.09i |
      .58+.98i .68+.58i  .45+.76i |
      .63+.55i .59+.97i  .95+.25i |
      .63+.91i .94+.52i  .17+.29i |
      .47+.47i .23+.92i  .7+.81i  |
      .75+.72i .44+.74i  .23+.36i |
      .1+.38i  .62+.93i  .39+.73i |
      .096+.3i .51+.96i  .43+.27i |

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

o14 = | .9+.62i    .57+.9i  |
      | .63+.3i    .22+.82i |
      | .75+.27i   .74+.82i |
      | .23+.59i   .94+.09i |
      | .72+.51i   .62+.37i |
      | .53+.31i   .43+.77i |
      | .58+.69i   .75+.59i |
      | .019+.061i .8+.5i   |
      | .76+.04i   .82+.18i |
      | .32+.68i   .12+.72i |

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

o15 = | .19-.43i  1.2+.98i  |
      | .52+1.2i  .05+1.4i  |
      | .3-.39i   .67-.12i  |
      | .57+.37i  1.1-.4i   |
      | -.17-.51i .3-.24i   |
      | .55-.53i  .72-.33i  |
      | -.07-.67i -1.4+.07i |
      | -.68+1.8i -.11-.18i |
      | .44+.52i  -.25+.25i |
      | -.3-1.2i  -1.3-.76i |

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

o16 = 1.0299525296675e-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 = | .87  .91 .71  .4  .51 |
      | .17  .79 .76  .15 .65 |
      | .3   .37 .091 .6  .93 |
      | .028 .73 .76  .65 .76 |
      | .64  .95 .11  .7  .9  |

                 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 = | 1.5  -.48  .91  -.89 -.67 |
      | -.98 .75   -2.1 .13  2.1  |
      | .98  -.069 .51  .63  -1.6 |
      | .38  -2.3  -.41 2    .2   |
      | -.41 1.4   1.9  -1.1 -.6  |

                 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 = 8.88178419700125e-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 = | 1.5  -.48  .91  -.89 -.67 |
      | -.98 .75   -2.1 .13  2.1  |
      | .98  -.069 .51  .63  -1.6 |
      | .38  -2.3  -.41 2    .2   |
      | -.41 1.4   1.9  -1.1 -.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 :