Next: Iterative Techniques, Previous: Basics, Up: Sparse Matrices [Contents][Index]
Octave includes a polymorphic solver for sparse matrices, where the exact solver used to factorize the matrix, depends on the properties of the sparse matrix itself. Generally, the cost of determining the matrix type is small relative to the cost of factorizing the matrix itself, but in any case the matrix type is cached once it is calculated, so that it is not re-determined each time it is used in a linear equation.
The selection tree for how the linear equation is solve is
spparms ("bandden")
continue, else goto 4.
The band density is defined as the number of nonzero values in the band
divided by the total number of values in the full band. The banded
matrix solvers can be entirely disabled by using spparms to set
bandden
to 1 (i.e., spparms ("bandden", 1)
).
The QR solver factorizes the problem with a Dulmage-Mendelsohn decomposition, to separate the problem into blocks that can be treated as over-determined, multiple well determined blocks, and a final over-determined block. For matrices with blocks of strongly connected nodes this is a big win as LU decomposition can be used for many blocks. It also significantly improves the chance of finding a solution to over-determined problems rather than just returning a vector of NaN’s.
All of the solvers above, can calculate an estimate of the condition number. This can be used to detect numerical stability problems in the solution and force a minimum norm solution to be used. However, for narrow banded, triangular or diagonal matrices, the cost of calculating the condition number is significant, and can in fact exceed the cost of factoring the matrix. Therefore the condition number is not calculated in these cases, and Octave relies on simpler techniques to detect singular matrices or the underlying LAPACK code in the case of banded matrices.
The user can force the type of the matrix with the matrix_type
function. This overcomes the cost of discovering the type of the matrix.
However, it should be noted that identifying the type of the matrix incorrectly
will lead to unpredictable results, and so matrix_type
should be
used with care.
Estimate the 2-norm of the matrix A using a power series analysis.
This is typically used for large matrices, where the cost of calculating
norm (A)
is prohibitive and an approximation to the 2-norm is
acceptable.
tol is the tolerance to which the 2-norm is calculated. By default tol is 1e-6.
The optional output iter returns the number of iterations that were
required for normest
to converge.
Estimate the 1-norm of the matrix A using a block algorithm.
normest1
is best for large sparse matrices where only an estimate of
the norm is required. For small to medium sized matrices, consider using
norm (A, 1)
. In addition, normest1
can be used for the
estimate of the 1-norm of a linear operator A when matrix-vector
products A * x
and A' * x
can be
cheaply computed. In this case, instead of the matrix A, a function
Afun (flag, x)
is used; it must return:
"dim"
"real"
A * x
, if flag is "notransp"
A' * x
, if flag is "transp"
A typical case is A defined by b ^ m
, in which the
result A * x
can be computed without even forming
explicitly b ^ m
by:
y = x; for i = 1:m y = b * y; endfor
The parameters p1, p2, … are arguments of
Afun (flag, x, p1, p2, …)
.
The default value for t is 2. The algorithm requires matrix-matrix products with sizes n x n and n x t.
The initial matrix x0 should have columns of unit 1-norm. The default
initial matrix x0 has the first column
ones (n, 1) / n
and, if t > 1, the remaining
columns with random elements -1 / n
, 1 / n
,
divided by n.
On output, nest is the desired estimate, v and w
are vectors such that w = A * v
, with
norm (w, 1)
= c * norm (v, 1)
. iter
contains in iter(1)
the number of iterations (the maximum is
hardcoded to 5) and in iter(2)
the total number of products
A * x
or A' * x
performed by the
algorithm.
Algorithm Note: normest1
uses random numbers during evaluation.
Therefore, if consistent results are required, the "state"
of the
random generator should be fixed before invoking normest1
.
Reference: N. J. Higham and F. Tisseur, A block algorithm for matrix 1-norm estimation, with and application to 1-norm pseudospectra, SIAM J. Matrix Anal. Appl., pp. 1185–1201, Vol 21, No. 4, 2000.
Estimate the 1-norm condition number of a square matrix A using t test vectors and a randomized 1-norm estimator.
The optional input t specifies the number of test vectors (default 5).
If the matrix is not explicit, e.g., when estimating the condition number of
A given an LU factorization, condest
uses the following
functions:
"dim"
"real"
a * x
, if flag is "notransp"
a' * x
, if flag is "transp"
"dim"
"real"
a \ x
, if flag is "notransp"
a' \ x
, if flag is "transp"
The parameters p1, p2, … are arguments of
Afcn (flag, x, p1, p2, …)
and solvefcn (flag, x, p1, p2,
…)
.
The principal output is the 1-norm condition number estimate cest.
The optional second output is an approximate null vector when cest is
large; it satisfies the equation
norm (A*v, 1) == norm (A, 1) * norm (v, 1) / est
.
Algorithm Note: condest
uses a randomized algorithm to approximate
the 1-norms. Therefore, if consistent results are required, the
"state"
of the random generator should be fixed before invoking
condest
.
References:
Query or set the parameters used by the sparse solvers and factorization functions.
The first four calls above get information about the current settings, while the others change the current settings. The parameters are stored as pairs of keys and values, where the values are all floats and the keys are one of the following strings:
Printing level of debugging information of the solvers (default 0)
Included for compatibility. Not used. (default 1)
Included for compatibility. Not used. (default 1)
Included for compatibility. Not used. (default 0)
Included for compatibility. Not used. (default 3)
Included for compatibility. Not used. (default 3)
Included for compatibility. Not used. (default 0.5)
Flag whether the LU/QR and the ’\’ and ’/’ operators will automatically use the sparsity preserving mmd functions (default 1)
Flag whether the LU and the ’\’ and ’/’ operators will automatically use the sparsity preserving amd functions (default 1)
The pivot tolerance of the UMFPACK solvers (default 0.1)
The pivot tolerance of the UMFPACK symmetric solvers (default 0.001)
The density of nonzero elements in a banded matrix before it is treated by the LAPACK banded solvers (default 0.5)
Flag whether the UMFPACK or mmd solvers are used for the LU, ’\’ and ’/’ operations (default 1)
The value of individual keys can be set with
spparms (key, val)
.
The default values can be restored with the special keyword
"default"
. The special keyword "tight"
can be used to
set the mmd solvers to attempt a sparser solution at the potential cost of
longer running time.
Calculate the structural rank of the sparse matrix S.
Note that only the structure of the matrix is used in this calculation based
on a Dulmage-Mendelsohn permutation to block triangular form. As
such the numerical rank of the matrix S is bounded by
sprank (S) >= rank (S)
. Ignoring floating point errors
sprank (S) == rank (S)
.
See also: dmperm.
Perform a symbolic factorization analysis of the sparse matrix S.
The input variables are
S is a real or complex sparse matrix.
Is the type of the factorization and can be one of
"sym"
(default)Factorize S. Assumes S is symmetric and uses the upper triangular portion of the matrix.
"col"
Factorize S' * S
.
"row"
Factorize S * S'
.
"lo"
Factorize S'
. Assumes S is symmetric and uses the lower
triangular portion of the matrix.
When mode is unspecified return the Cholesky factorization for
R. If mode is "lower"
or "L"
then return
the conjugate transpose R'
which is a lower triangular factor.
The conjugate transpose version is faster and uses less memory, but still
returns the same values for all other outputs: count, h,
parent, and post.
The output variables are:
The row counts of the Cholesky factorization as determined by
typ. The computational difficulty of performing the true
factorization using chol
is sum (count .^ 2)
.
The height of the elimination tree.
The elimination tree itself.
A sparse boolean matrix whose structure is that of the Cholesky factorization as determined by typ.
See also: chol, etree, treelayout.
For non square matrices, the user can also utilize the spaugment
function to find a least squares solution to a linear equation.
Create the augmented matrix of A.
This is given by
[c * eye(m, m), A; A', zeros(n, n)]
This is related to the least squares solution of
A \ b
, by
s * [ r / c; x] = [ b, zeros(n, columns(b)) ]
where r is the residual error
r = b - A * x
As the matrix s is symmetric indefinite it can be factorized with
lu
, and the minimum norm solution can therefore be found without the
need for a qr
factorization. As the residual error will be
zeros (m, m)
for underdetermined problems, and example
can be
m = 11; n = 10; mn = max (m, n); A = spdiags ([ones(mn,1), 10*ones(mn,1), -ones(mn,1)], [-1, 0, 1], m, n); x0 = A \ ones (m,1); s = spaugment (A); [L, U, P, Q] = lu (s); x1 = Q * (U \ (L \ (P * [ones(m,1); zeros(n,1)]))); x1 = x1(end - n + 1 : end);
To find the solution of an overdetermined problem needs an estimate of the
residual error r and so it is more complex to formulate a minimum norm
solution using the spaugment
function.
In general the left division operator is more stable and faster than using
the spaugment
function.
See also: mldivide.
Finally, the function eigs
can be used to calculate a limited
number of eigenvalues and eigenvectors based on a selection criteria
and likewise for svds
which calculates a limited number of
singular values and vectors.
Calculate a limited number of eigenvalues and eigenvectors of A, based on a selection criteria.
The number of eigenvalues and eigenvectors to calculate is given by k and defaults to 6.
By default, eigs
solve the equation
where
is the corresponding eigenvector. If given the positive definite matrix
B then eigs
solves the general eigenvalue equation
The argument sigma determines which eigenvalues are returned. sigma can be either a scalar or a string. When sigma is a scalar, the k eigenvalues closest to sigma are returned. If sigma is a string, it must have one of the following values.
"lm"
Largest Magnitude (default).
"sm"
Smallest Magnitude.
"la"
Largest Algebraic (valid only for real symmetric problems).
"sa"
Smallest Algebraic (valid only for real symmetric problems).
"be"
Both Ends, with one more from the high-end if k is odd (valid only for real symmetric problems).
"lr"
Largest Real part (valid only for complex or unsymmetric problems).
"sr"
Smallest Real part (valid only for complex or unsymmetric problems).
"li"
Largest Imaginary part (valid only for complex or unsymmetric problems).
"si"
Smallest Imaginary part (valid only for complex or unsymmetric problems).
If opts is given, it is a structure defining possible options that
eigs
should use. The fields of the opts structure are:
issym
If af is given, then flags whether the function af defines a symmetric problem. It is ignored if A is given. The default is false.
isreal
If af is given, then flags whether the function af defines a real problem. It is ignored if A is given. The default is true.
tol
Defines the required convergence tolerance, calculated as
tol * norm (A)
. The default is eps
.
maxit
The maximum number of iterations. The default is 300.
p
The number of Lanzcos basis vectors to use. More vectors will result in
faster convergence, but a greater use of memory. The optimal value of
p
is problem dependent and should be in the range k + 1
to n. The default value is 2 * k
.
v0
The starting vector for the algorithm. An initial vector close to the
final vector will speed up convergence. The default is for ARPACK
to randomly generate a starting vector. If specified, v0
must be
an n-by-1 vector where n = rows (A)
disp
The level of diagnostic printout (0|1|2). If disp
is 0 then
diagnostics are disabled. The default value is 0.
cholB
Flag if chol (B)
is passed rather than B. The default is
false.
permB
The permutation vector of the Cholesky factorization of B if
cholB
is true. It is obtained by [R, ~, permB] =
chol (B,
. The default is "vector"
)1:n
.
It is also possible to represent A by a function denoted af. af must be followed by a scalar argument n defining the length of the vector argument accepted by af. af can be a function handle, an inline function, or a string. When af is a string it holds the name of the function to use.
af is a function of the form y = af (x)
where the required
return value of af is determined by the value of sigma. The
four possible forms are
A * x
if sigma is not given or is a string other than "sm".
A \ x
if sigma is 0 or "sm".
(A - sigma * I) \ x
for the standard eigenvalue problem, where I
is the identity matrix
of the same size as A.
(A - sigma * B) \ x
for the general eigenvalue problem.
The return arguments of eigs
depend on the number of return arguments
requested. With a single return argument, a vector d of length
k is returned containing the k eigenvalues that have been
found. With two return arguments, V is a n-by-k matrix
whose columns are the k eigenvectors corresponding to the returned
eigenvalues. The eigenvalues themselves are returned in d in the
form of a n-by-k matrix, where the elements on the diagonal
are the eigenvalues.
Given a third return argument flag, eigs
returns the status
of the convergence. If flag is 0 then all eigenvalues have converged.
Any other value indicates a failure to converge.
This function is based on the ARPACK package, written by R. Lehoucq, K. Maschhoff, D. Sorensen, and C. Yang. For more information see http://www.caam.rice.edu/software/ARPACK/.
Find a few singular values of the matrix A.
The singular values are calculated using
[m, n] = size (A); s = eigs ([sparse(m, m), A; A', sparse(n, n)])
The eigenvalues returned by eigs
correspond to the singular values
of A. The number of singular values to calculate is given by k
and defaults to 6.
The argument sigma specifies which singular values to find. When
sigma is the string 'L'
, the default, the largest singular
values of A are found. Otherwise, sigma must be a real scalar
and the singular values closest to sigma are found. As a corollary,
sigma = 0
finds the smallest singular values. Note that for
relatively small values of sigma, there is a chance that the
requested number of singular values will not be found. In that case
sigma should be increased.
opts is a structure defining options that svds
will pass
to eigs
. The possible fields of this structure are documented in
eigs
. By default, svds
sets the following three fields:
tol
The required convergence tolerance for the singular values. The default
value is 1e-10. eigs
is passed tol / sqrt(2)
.
maxit
The maximum number of iterations. The default is 300.
disp
The level of diagnostic printout (0|1|2). If disp
is 0 then
diagnostics are disabled. The default value is 0.
If more than one output is requested then svds
will return an
approximation of the singular value decomposition of A
A_approx = u*s*v'
where A_approx is a matrix of size A but only rank k.
flag returns 0 if the algorithm has successfully converged, and 1 otherwise. The test for convergence is
norm (A*v - u*s, 1) <= tol * norm (A, 1)
svds
is best for finding only a few singular values from a large
sparse matrix. Otherwise, svd (full (A))
will likely be more
efficient.
The CHOLMOD, UMFPACK and CXSPARSE packages were written by Tim Davis and are available at http://faculty.cse.tamu.edu/davis/suitesparse.html
Next: Iterative Techniques, Previous: Basics, Up: Sparse Matrices [Contents][Index]