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Package matrix (in matrix.i) - LAPACK linear algebra functions
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DOCUMENT LUrcond(a)
or LUrcond(a, one_norm=1)
returns the reciprocal condition number of the N-by-N matrix A.
If the ONE_NORM argument is non-nil and non-zero, the 1-norm
condition number is returned, otherwise the infinity-norm condition
number is returned.
The condition number is the ratio of the largest to the smallest
singular value, max(singular_values)*max(1/singular_values) (or
sum(abs(singular_values)*sum(abs(1/singular_values)) if ONE_NORM
is selected?). If the reciprocal condition number is near zero
then A is numerically singular; specifically, if
1.0+LUrcond(a) == 1.0
then A is numerically singular.
SEE ALSO: LUsolve
DOCUMENT LUsolve(a, b)
or LUsolve(a, b, which=which)
or a_inverse= LUsolve(a)
returns the solution x of the matrix equation:
A(,+)*x(+) = B
If A is an n-by-n matrix then B must have length n, and the returned
x will also have length n.
B may have additional dimensions, in which case the returned x
will have the same additional dimensions. The WHICH dimension of B,
and of the returned x is the one of length n which participates
in the matrix solve. By default, WHICH=1, so that the equations
being solved are:
A(,+)*x(+,..) = B
Non-positive WHICH counts from the final dimension (as for the
sort and transpose functions), so that WHICH=0 solves:
x(..,+)*A(,+) = B
Other examples:
A_ij X_jklm = B_iklm (WHICH=1)
A_ij X_kjlm = B_kilm (WHICH=2)
A_ij X_klmj = B_klmi (WHICH=4 or WHICH=0)
If the B argument is omitted, the inverse of A is returned:
A(,+)*x(+,) and A(,+)*x(,+) will be unit matrices.
LUsolve works by LU decomposition using Gaussian elimination with
pivoting. It is the fastest way to solve square matrices. QRsolve
handles non-square matrices, as does SVsolve. SVsolve is slowest,
but can deal with highly singular matrices sensibly.
DOCUMENT QRsolve(a, b)
or QRsolve(a, b, which=which)
returns the solution x (in a least squares or least norm sense
described below) of the matrix equation:
A(,+)*x(+) = B
If A is an m-by-n matrix (i.e.- m equations in n unknowns), then B
must have length m, and the returned x will have length n.
If nm, the system is underdetermined -- many solutions are possible
-- the returned x has minimum L2 norm among all solutions
B may have additional dimensions, in which case the returned x
will have the same additional dimensions also have those dimensions.
The WHICH dimension of B and the returned x is the one of length m
or n which participates in the matrix solve. By default, WHICH=1,
so that the equations being solved are:
A(,+)*x(+,..) = B
Non-positive WHICH counts from the final dimension (as for the
sort and transpose functions), so that WHICH=0 solves:
A(,+)*x(..,+) = B
QRsolve works by QR factorization if nm.
QRsolve is slower than LUsolve. Its main attraction is that it can
handle overdetermined or underdetermined systems of equations
(nonsquare matrices). QRsolve may fail for singular systems; try
SVsolve in this case.
DOCUMENT s= SVdec(a, u, vt)
or s= SVdec(a, u, vt, full=1)
performs the singular value decomposition of the m-by-n matrix A:
A = (U(,+) * SIGMA(+,))(,+) * VT(+,)
where U is an m-by-m orthogonal matrix, VT is an n-by-n orthogonal
matrix, and SIGMA is an m-by-n matrix which is zero except for its
min(m,n) diagonal elements. These diagonal elements are the return
value of the function, S. The returned S is always arranged in
order of descending absolute value. U(,1:min(m,n)) are the left
singular vectors corresponding to the min(m,n) elements of S;
VT(1:min(m,n),) are the right singular vectors. (The original A
matrix maps a right singular vector onto the corresponding left
singular vector, stretched by a factor of the singular value.)
Note that U and VT are strictly outputs; if you don't need them,
they need not be present in the calling sequence.
By default, U will be an m-by-min(m,n) matrix, and V will be
a min(m,n)-by-n matrix (i.e.- only the singular vextors are returned,
not the full orthogonal matrices). Set the FULL keyword to a
non-zero value to get the full m-by-m and n-by-n matrices.
On rare occasions, the routine may fail; if it does, the
first SVinfo values of the returned S are incorrect. Hence,
the external variable SVinfo will be 0 after a successful call
to SVdec. If SVinfo>0, then external SVe contains the superdiagonal
elements of the bidiagonal matrix whose diagonal is the returned
S, and that bidiagonal matrix is equal to (U(+,)*A(+,))(,+) * V(+,).
Numerical Recipes (Press, et. al. Cambridge University Press 1988)
has a good discussion of how to use the SVD -- see section 2.9.
DOCUMENT SVsolve(a, b)
or SVsolve(a, b, rcond)
or SVsolve(a, b, rcond, which=which)
returns the solution x (in a least squares sense described below) of
the matrix equation:
A(,+)*x(+) = B
If A is an m-by-n matrix (i.e.- m equations in n unknowns), then B
must have length m, and the returned x will have length n.
If nm, the system is underdetermined -- many solutions are possible
-- the returned x has minimum L2 norm among all solutions
SVsolve works by singular value decomposition, therefore it is
immune to failure due to singularity of the A matrix. The optional
RCOND argument defaults to 1.0e-9; singular values less than RCOND
times the largest singular value (absolute value) will be set to 0.0.
If RCOND<=0.0, machine precision is used. The effective rank of the
matrix is returned as the external variable SVrank.
You can examine the details of the SVD by calling the SVdec routine,
which returns the singular vectors as well as the singular values.
Numerical Recipes (Press, et. al. Cambridge University Press 1988)
has a good discussion of how to use the SVD -- see section 2.9.
B may have additional dimensions, in which case the returned x
will have the same additional dimensions. The WHICH argument
(default 1) controls which dimension of B takes part in the matrix
solve. See QRsolve or LUsolve for a complete discussion.
DOCUMENT TDsolve(c, d, e, b)
or TDsolve(c, d, e, b, which=which)
returns the solution to the tridiagonal system:
D(1)*x(1) + E(1)*x(2) = B(1)
C(1:-1)*x(1:-2) + D(2:-1)*x(2:-1) + E(2:0)*x(3:0) = B(2:-1)
C(0)*x(-1) + D(0)*x(0) = B(0)
(i.e.- C is the subdiagonal, D the diagonal, and E the superdiagonal;
C and E have one fewer element than D, which is the same length as
both B and x)
B may have additional dimensions, in which case the returned x
will have the same additional dimensions. The WHICH dimension of B,
and of the returned x is the one of length n which participates
in the matrix solve. By default, WHICH=1, so that the equations
being solved involve B(,..) and x(+,..).
Non-positive WHICH counts from the final dimension (as for the
sort and transpose functions), so that WHICH=0 involves B(..,)
and x(..,+).
The C, D, and E arguments may be either scalars or vectors; they
will be broadcast as appropriate.