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21.2 Linear Algebra on Sparse Matrices

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

  1. If the matrix is diagonal, solve directly and goto 8
  2. If the matrix is a permuted diagonal, solve directly taking into account the permutations. Goto 8
  3. If the matrix is square, banded and if the band density is less than that given by spparms ("bandden") continue, else goto 4.
    1. If the matrix is tridiagonal and the right-hand side is not sparse continue, else goto 3b.
      1. If the matrix is hermitian, with a positive real diagonal, attempt Cholesky factorization using LAPACK xPTSV.
      2. If the above failed or the matrix is not hermitian with a positive real diagonal use Gaussian elimination with pivoting using LAPACK xGTSV, and goto 8.
    2. If the matrix is hermitian with a positive real diagonal, attempt Cholesky factorization using LAPACK xPBTRF.
    3. if the above failed or the matrix is not hermitian with a positive real diagonal use Gaussian elimination with pivoting using LAPACK xGBTRF, and goto 8.
  4. If the matrix is upper or lower triangular perform a sparse forward or backward substitution, and goto 8
  5. If the matrix is a upper triangular matrix with column permutations or lower triangular matrix with row permutations, perform a sparse forward or backward substitution, and goto 8
  6. If the matrix is square, hermitian with a real positive diagonal, attempt sparse Cholesky factorization using CHOLMOD.
  7. If the sparse Cholesky factorization failed or the matrix is not hermitian with a real positive diagonal, and the matrix is square, factorize using UMFPACK.
  8. If the matrix is not square, or any of the previous solvers flags a singular or near singular matrix, find a minimum norm solution using CXSPARSE(9).

The band density is defined as the number of non-zero values in the matrix divided by the number of non-zero values in the matrix. 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, 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.

Function File: [n, c] = normest (a, tol)

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 the 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. c returns the number of iterations needed for normest to converge.

Function File: [est, v, w, iter] = onenormest (a, t)
Function File: [est, v, w, iter] = onenormest (apply, apply_t, n, t)

Apply Higham and Tisseur's randomized block 1-norm estimator to matrix a using t test vectors. If t exceeds 5, then only 5 test vectors are used.

If the matrix is not explicit, e.g., when estimating the norm of inv (A) given an LU factorization, onenormest applies A and its conjugate transpose through a pair of functions apply and apply_t, respectively, to a dense matrix of size n by t. The implicit version requires an explicit dimension n.

Returns the norm estimate est, two vectors v and w related by norm (w, 1) = est * norm (v, 1), and the number of iterations iter. The number of iterations is limited to 10 and is at least 2.

References:

See also: condest, norm, cond.

Function File: [est, v] = condest (a, t)
Function File: [est, v] = condest (a, solve, solve_t, t)
Function File: [est, v] = condest (apply, apply_t, solve, solve_t, n, t)

Estimate the 1-norm condition number of a matrix A using t test vectors using a randomized 1-norm estimator. If t exceeds 5, then only 5 test vectors are used.

If the matrix is not explicit, e.g., when estimating the condition number of a given an LU factorization, condest uses the following functions:

apply

A*x for a matrix x of size n by t.

apply_t

A'*x for a matrix x of size n by t.

solve

A \ b for a matrix b of size n by t.

solve_t

A' \ b for a matrix b of size n by t.

The implicit version requires an explicit dimension n.

condest uses a randomized algorithm to approximate the 1-norms.

condest returns the 1-norm condition estimate est and a vector v satisfying norm (A*v, 1) == norm (A, 1) * norm (v, 1) / est. When est is large, v is an approximate null vector.

References:

See also: cond, norm, onenormest.

Loadable Function: spparms ()
Loadable Function: vals = spparms ()
Loadable Function: [keys, vals] = spparms ()
Loadable Function: val = spparms (key)
Loadable Function: spparms (vals)
Loadable Function: spparms ('defaults')
Loadable Function: spparms ('tight')
Loadable Function: spparms (key, val)

Sets or displays 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:

spumoni

Printing level of debugging information of the solvers (default 0)

ths_rel

Included for compatibility. Not used. (default 1)

ths_abs

Included for compatibility. Not used. (default 1)

exact_d

Included for compatibility. Not used. (default 0)

supernd

Included for compatibility. Not used. (default 3)

rreduce

Included for compatibility. Not used. (default 3)

wh_frac

Included for compatibility. Not used. (default 0.5)

autommd

Flag whether the LU/QR and the '\' and '/' operators will automatically use the sparsity preserving mmd functions (default 1)

autoamd

Flag whether the LU and the '\' and '/' operators will automatically use the sparsity preserving amd functions (default 1)

piv_tol

The pivot tolerance of the UMFPACK solvers (default 0.1)

sym_tol

The pivot tolerance of the UMFPACK symmetric solvers (default 0.001)

bandden

The density of non-zero elements in a banded matrix before it is treated by the LAPACK banded solvers (default 0.5)

umfpack

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 'defaults'. The special keyword 'tight' can be used to set the mmd solvers to attempt for a sparser solution at the potential cost of longer running time.

Loadable Function: p = sprank (s)

Calculates the structural rank of a 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.

Loadable Function: [count, h, parent, post, r] = symbfact (s, typ, mode)

Performs a symbolic factorization analysis on the sparse matrix s. Where

s

s is a complex or real sparse matrix.

typ

Is the type of the factorization and can be one of

sym

Factorize s. This is the default.

col

Factorize s' * s.

row

Factorize s * s'.

lo

Factorize s'

mode

The default is to return the Cholesky factorization for r, and if mode is 'L', the conjugate transpose of the Cholesky factorization is returned. The conjugate transpose version is faster and uses less memory, but returns the same values for count, h, parent and post outputs.

The output variables are

count

The row counts of the Cholesky factorization as determined by typ.

h

The height of the elimination tree.

parent

The elimination tree itself.

post

A sparse boolean matrix whose structure is that of the Cholesky factorization as determined by typ.

For non square matrices, the user can also utilize the spaugment function to find a least squares solution to a linear equation.

Function File: s = spaugment (a, c)

Creates 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 under determined 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.

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.

Loadable Function: d = eigs (a)
Loadable Function: d = eigs (a, k)
Loadable Function: d = eigs (a, k, sigma)
Loadable Function: d = eigs (a, k, sigma,opts)
Loadable Function: d = eigs (a, b)
Loadable Function: d = eigs (a, b, k)
Loadable Function: d = eigs (a, b, k, sigma)
Loadable Function: d = eigs (a, b, k, sigma, opts)
Loadable Function: d = eigs (af, n)
Loadable Function: d = eigs (af, n, b)
Loadable Function: d = eigs (af, n, k)
Loadable Function: d = eigs (af, n, b, k)
Loadable Function: d = eigs (af, n, k, sigma)
Loadable Function: d = eigs (af, n, b, k, sigma)
Loadable Function: d = eigs (af, n, k, sigma, opts)
Loadable Function: d = eigs (af, n, b, k, sigma, opts)
Loadable Function: [v, d] = eigs (a, …)
Loadable Function: [v, d] = eigs (af, n, …)
Loadable Function: [v, d, flag] = eigs (a, …)
Loadable Function: [v, d, flag] = eigs (af, n, …)

Calculate a limited number of eigenvalues and eigenvectors of a, based on a selection criteria. The number eigenvalues and eigenvectors to calculate is given by k whose default value is 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 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 some of the options that eigs should use. The fields of the structure opts 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, given 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 larger amount of memory. The optimal value of 'p' is problem dependent and should be in the range k to n. The default value is 2 * k.

v0

The starting vector for the computation. The default is to have ARPACK randomly generate a starting vector.

disp

The level of diagnostic printout. If disp is 0 then there is no printout. The default value is 1.

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. That is chol ( b (permB, permB)). The default is 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 passed either as an inline function, function handle or as a string. In the case where af is passed as a string, the name of the string defines the function to use.

af is a function of the form function y = af (x), y = …; endfunction, where the required return value of af is determined by the value of sigma, and are

A * x

If sigma is not given or is a string other than 'sm'.

A \ x

If sigma is 'sm'.

(A - sigma * I) \ x

for standard eigenvalue problem, where I is the identity matrix of the same size as A. If sigma is zero, this reduces the A \ x.

(A - sigma * B) \ x

for the general eigenvalue problem.

The return arguments of eigs depends on the number of return arguments. With a single return argument, a vector d of length k is returned, represent 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 then 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 also returns the status of the convergence. If flag is 0, then all eigenvalues have converged, otherwise not.

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/.

See also: eig, svds.

Function File: s = svds (a)
Function File: s = svds (a, k)
Function File: s = svds (a, k, sigma)
Function File: s = svds (a, k, sigma, opts)
Function File: [u, s, v, flag] = svds (…)

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, whose default value is 6.

The argument sigma can be used to specify which singular values to find. sigma can be either the string 'L', the default, in which case the largest singular values of a are found. Otherwise sigma should be a real scalar, in which case the singular values closest to sigma are found. Note that for relatively small values of sigma, there is the chance that the requested number of singular values are not returned. In that case sigma should be increased.

If opts is given, then it is a structure that defines options that svds will pass to eigs. The possible fields of this structure are therefore determined by eigs. By default three fields of this structure are set by svds.

tol

The required convergence tolerance for the singular values. eigs is passed tol divided by sqrt(2). The default value is 1e-10.

maxit

The maximum number of iterations. The default is 300.

disp

The level of diagnostic printout. If disp is 0 then there is no printout. The default value is 0.

If more than one output argument is given, then svds also calculates the left and right singular vectors of a. flag is used to signal the convergence of svds. If svds converges to the desired tolerance, then flag given by

 
norm (a * v - u * s, 1) <= ...
        tol * norm (a, 1)

will be zero.

See also: eigs.

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