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doc ///
Key
solve
(solve,Matrix,Matrix)
(solve,MutableMatrix,MutableMatrix)
[solve,MaximalRank]
[solve,ClosestFit]
[solve,Precision]
[solve, Invertible] -- TODO: document this
Headline
solve linear equation(s)
Usage
X = solve(A,B)
Inputs
A:Matrix
or @ofClass MutableMatrix@, of size m by n over a field R, which can be
one of: @demark_", " \\ TT \ {"ZZ/p", "GF(p^n)", "QQ", "RR", "CC"}@
B:
of the same type of matrix as {\tt A}, over the same ring, of size m by r
ClosestFit => Boolean
whether to use the least squares method, in the case when the ring is {\tt RR} or {\tt CC}
MaximalRank => Boolean
declares to the system that the matrix is full rank. In some cases, this can dramatically
speed up the computation. If the matrix is not full rank, then the results are potentially
meaningless.
Outputs
X:
of the same type of matrix as {\tt A}, over the same ring, such that $AX=B$
Description
Example
kk = ZZ/101;
A = matrix"1,2,3,4;1,3,6,10;19,7,11,13" ** kk
b = matrix"1;1;1" ** kk
x = solve(A,b)
A*x-b
Example
kk = GF(25)
a = kk_0
A = matrix"a,a+1,a+2,3a,4;a-1,1,2a,6,10;19,7,a,11,13" ** kk
b = matrix"1;-a+1;1" ** kk
x = solve(A,b)
A*x-b
Example
kk = QQ
A = matrix"1,2,3,4;1,3,6,10;19,7,11,13" ** kk
b = matrix"1;1;1" ** kk
x = solve(A,b)
A*x-b
Text
Over {\tt RR_{53}} or {\tt CC_{53}}, if the matrix A is non-singular and square,
then highly optimized LAPACK routines will be called.
Example
printingPrecision = 4;
A = matrix "1,2,3;1,3,6;19,7,11" ** RR
b = matrix "1;1;1" ** RR
x = solve(A,b)
A*x-b
norm oo
clean(1e-15, A*x-b)
Text
If you know that your matrix is square, and invertible, then providing the
hint: MaximalRank=>true allows Macaulay2 to choose the fastest routines. For
small matrix sizes, it should not be too noticeable, but for large matrices, the
difference in time taken can be dramatic.
Example
printingPrecision = 4;
N = 40
A = mutableMatrix(CC_53, N, N); fillMatrix A;
B = mutableMatrix(CC_53, N, 2); fillMatrix B;
time X = solve(A,B);
time X = solve(A,B, MaximalRank=>true);
norm(A*X-B)
Text
Over higher precision RR or CC, these routines will be much slower than
the lower precision LAPACK routines.
Example
N = 100
A = mutableMatrix(CC_100, N, N); fillMatrix A;
B = mutableMatrix(CC_100, N, 2); fillMatrix B;
time X = solve(A,B);
time X = solve(A,B, MaximalRank=>true);
norm(A*X-B)
Text
Giving the option ClosestFit=>true, in the case when the field is RR or CC,
uses a least squares algorithm to find a best fit solution.
Example
kk = RR_53;
A = matrix"1,2,3,4;1,3,6,10;19,7,11,13" ** kk
b = matrix"1;1;1" ** kk
x1 = solve(A,b, ClosestFit=>true)
A*x1-b
Text
Giving both options ClosestFit and MaximalRank allows Macaulay2 to call
a faster algorithm.
Example
x2 = solve(A,b, ClosestFit=>true, MaximalRank=>true)
A*x2-b
Caveat
(1) This function is limited in scope, but has been designed to be much faster than
generic algorithms.
(2) If the matrix is a square invertible matrix, giving the option MaximalRank=>true can
strongly speed up the computation.
(3) For mutable matrices, this function is only currently implemented for densely encoded matrices.
SeeAlso
LUdecomposition
inverse
determinant
SVD
norm
clean
///
-*
document {
Key => {
solve,(solve,Matrix,Matrix),
(solve,MutableMatrix,MutableMatrix),
[solve,MaximalRank],
[solve,ClosestFit]
},
Headline => "solve a linear equation",
Usage => "x = solve(A,b)",
Inputs => {
"A" => {ofClass Matrix, ", or ", ofClass MutableMatrix, " of size m by n over either
a finite field ZZ/p, RR or CC"},
"b" => {"the same type of matrix, over the same ring, of size m by r"},
ClosestFit => Boolean => {"whether to use the least squares method"},
MaximalRank => Boolean => {"whether to assume the matrix has maximal rank, in case the least squares method is used"}
},
Outputs => {
"x" => {"the same type of matrix, over the same ring, of size n by r,
such that ", TT "Ax=b"}
},
PARA {
"(Disambiguation: for division of matrices, which can also be thought of as solving a
system of linear equations, see instead ", TO (symbol //,Matrix, Matrix), ". For lifting a map between modules to a map
between their free resolutions, see ", TO 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 ", TO "RR", " or ", TO "CC",
", the matrix ", TT "A", " must be a square
non-singular matrix. Third, if ", TT "A", " and ", TT "b",
" are mutable matrices over ", TO "RR", " or ", TO "CC", ", they must be dense matrices.",
EXAMPLE lines ///
kk = ZZ/101;
A = matrix"1,2,3,4;1,3,6,10;19,7,11,13" ** kk
b = matrix"1;1;1" ** kk
x = solve(A,b)
A*x-b
///,
"Over ", TO "RR", " or ", TO "CC", ", the matrix ", TT "A", " must be a non-singular square matrix.",
EXAMPLE lines ///
printingPrecision = 2;
A = matrix "1,2,3;1,3,6;19,7,11" ** RR
b = matrix "1;1;1" ** RR
x = solve(A,b)
A*x-b
norm oo
///,
"For large dense matrices over ", TO "RR", " or ", TO "CC", ", this function calls
the LAPACK routines.",
EXAMPLE lines ///
n = 10;
A = random(CC^n,CC^n)
b = random(CC^n,CC^2)
x = solve(A,b)
norm ( matrix A * matrix x - matrix b )
///,
"This may be used to invert a matrix over ", TT "ZZ/p", ", ", TO "RR", " or ", TT "QQ", ".",
EXAMPLE lines ///
A = random(RR^5, RR^5)
I = id_(target A)
A' = solve(A,I)
norm(A*A' - I)
norm(A'*A - I)
///,
"Another method, which isn't generally as fast, and isn't as stable over ", TO "RR", " or ", TO "CC", ",
is to lift the matrix ", TT "b", "
along the matrix ", TT "A", " (see ", TO (symbol//,Matrix,Matrix), ").",
EXAMPLE lines ///
A'' = I // A
norm(A' - A'')
///,
Caveat => {"This function is limited in scope, but is sometimes useful for very large
matrices"},
SeeAlso => {LUdecomposition, SVD, MutableMatrix, norm, random}
}
*-
|
