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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

- If the matrix is diagonal, solve directly and goto 8
- If the matrix is a permuted diagonal, solve directly taking into account the permutations. Goto 8
- If the matrix is square, banded and if the band density is less
than that given by
`spparms ("bandden")`

continue, else goto 4.- If the matrix is tridiagonal and the right-hand side is not sparse
continue, else goto 3b.
- If the matrix is Hermitian, with a positive real diagonal, attempt Cholesky factorization using LAPACK xPTSV.
- 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.

- If the matrix is Hermitian with a positive real diagonal, attempt Cholesky factorization using LAPACK xPBTRF.
- 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.

- If the matrix is tridiagonal and the right-hand side is not sparse
continue, else goto 3b.
- If the matrix is upper or lower triangular perform a sparse forward or backward substitution, and goto 8
- If the matrix is an upper triangular matrix with column permutations or lower triangular matrix with row permutations, perform a sparse forward or backward substitution, and goto 8
- If the matrix is square, Hermitian with a real positive diagonal, attempt sparse Cholesky factorization using CHOLMOD.
- If the sparse Cholesky factorization failed or the matrix is not Hermitian with a real positive diagonal, and the matrix is square, factorize, solve, and perform one refinement iteration using UMFPACK.
- 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
^{10}.

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.

- :
`nest`=**normest***(*¶`A`) - :
`nest`=**normest***(*¶`A`,`tol`) - :
*[*`nest`,`iter`] =**normest***(…)*¶ 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 (`

is prohibitive and an approximation to the 2-norm is acceptable.`A`)`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.

- :
`nest`=**normest1***(*¶`A`) - :
`nest`=**normest1***(*¶`A`,`t`) - :
`nest`=**normest1***(*¶`A`,`t`,`x0`) - :
`nest`=**normest1***(*¶`Afcn`,`t`,`x0`,`p1`,`p2`, …) - :
*[*`nest`,`v`] =**normest1***(*¶`A`, …) - :
*[*`nest`,`v`,`w`] =**normest1***(*¶`A`, …) - :
*[*`nest`,`v`,`w`,`iter`] =**normest1***(*¶`A`, …) 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 (`

. In addition,`A`, 1)`normest1`

can be used for the estimate of the 1-norm of a linear operator`A`when matrix-vector products

and`A`*`x`

can be cheaply computed. In this case, instead of the matrix`A`' *`x``A`, a function

is used; it must return:`Afcn`(`flag`,`x`)- the dimension
`n`of`A`, if`flag`is`"dim"`

- true if
`A`is a real operator, if`flag`is`"real"`

- the result

, if`A`*`x``flag`is`"notransp"`

- the result

, if`A`' *`x``flag`is`"transp"`

A typical case is

`A`defined by

, in which the result`b`^`m`

can be computed without even forming explicitly`A`*`x`

by:`b`^`m``y`=`x`; for`i`= 1:`m``y`=`b`*`y`; endforThe parameters

`p1`,`p2`, … are arguments of

.`Afcn`(`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 (`

and, if`n`, 1) /`n``t`> 1, the remaining columns with random elements`-1 /`

,`n``1 /`

, divided by`n``n`.On output,

`nest`is the desired estimate,`v`and`w`are vectors such that

, with`w`=`A`*`v``norm (`

=`w`, 1)

.`c`* norm (`v`, 1)`iter`contains in

the number of iterations (the maximum is hardcoded to 5) and in`iter`(1)

the total number of products`iter`(2)

or`A`*`x`

performed by the algorithm.`A`' *`x`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.

- the dimension

- :
`cest`=**condest***(*¶`A`) - :
`cest`=**condest***(*¶`A`,`t`) - :
`cest`=**condest***(*¶`A`,`Ainvfcn`) - :
`cest`=**condest***(*¶`A`,`Ainvfcn`,`t`) - :
`cest`=**condest***(*¶`A`,`Ainvfcn`,`t`,`p1`,`p2`, …) - :
`cest`=**condest***(*¶`Afcn`,`Ainvfcn`) - :
`cest`=**condest***(*¶`Afcn`,`Ainvfcn`,`t`) - :
`cest`=**condest***(*¶`Afcn`,`Ainvfcn`,`t`,`p1`,`p2`, …) - :
*[*`cest`,`v`] =**condest***(…)*¶ -
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).The input may be a matrix

`A`(the algorithm is particularly appropriate for large, sparse matrices). Alternatively, the behavior of the matrix can be defined implicitly by functions. When using an implicit definition,`condest`

requires the following functions:- -

which must return`Afcn`(`flag`,`x`)- the dimension
`n`of`A`, if`flag`is`"dim"`

- true if
`A`is a real operator, if`flag`is`"real"`

- the result

, if`A`*`x``flag`is "notransp" - the result

, if`A`' *`x``flag`is "transp"

- the dimension
- -

which must return`Ainvfcn`(`flag`,`x`)- the dimension
`n`of`inv (`

, if`A`)`flag`is`"dim"`

- true if
`inv (`

is a real operator, if`A`)`flag`is`"real"`

- the result
`inv (`

, if`A`) *`x``flag`is "notransp" - the result
`inv (`

, if`A`)' *`x``flag`is "transp"

- the dimension

Any parameters

`p1`,`p2`, … are additional arguments of

and`Afcn`(`flag`,`x`,`p1`,`p2`, …)

.`Ainvfcn`(`flag`,`x`,`p1`,`p2`, …)The principal output is the 1-norm condition number estimate

`cest`.The optional second output

`v`is an approximate null vector; it satisfies the equation`norm (`

.`A`*`v`, 1) == norm (`A`, 1) * norm (`v`, 1) /`cest`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:

- N.J. Higham and F. Tisseur, A Block Algorithm for Matrix 1-Norm Estimation, with an Application to 1-Norm Pseudospectra. SIMAX vol 21, no 4, pp 1185–1201. https://dx.doi.org/10.1137/S0895479899356080
- N.J. Higham and F. Tisseur, A Block Algorithm for Matrix 1-Norm Estimation, with an Application to 1-Norm Pseudospectra. https://citeseer.ist.psu.edu/223007.html

- -

- :
**spparms***()*¶ - :
`vals`=**spparms***()*¶ - :
*[*`keys`,`vals`] =**spparms***()*¶ - :
`val`=**spparms***(*¶`key`) - :
**spparms***(*¶`vals`) - :
**spparms***("default")*¶ - :
**spparms***("tight")*¶ - :
**spparms***(*¶`key`,`val`) 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:

- ‘
`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 nonzero 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 (`

. The default values can be restored with the special keyword`key`,`val`)`"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.- ‘

- :
`p`=**sprank***(*¶`S`) -
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 (`

. Ignoring floating point errors`S`) >= rank (`S`)`sprank (`

.`S`) == rank (`S`)**See also:**dmperm.

- :
*[*`count`,`h`,`parent`,`post`,`R`] =**symbfact***(*¶`S`) - :
*[…] =***symbfact***(*¶`S`,`typ`) - :
*[…] =***symbfact***(*¶`S`,`typ`,`mode`) -
Perform a symbolic factorization analysis of the sparse matrix

`S`.The input variables are

`S``S`is a real or complex sparse matrix.`typ`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

. Assumes`S`'`S`is symmetric and uses the lower triangular portion of the matrix.

`mode`When

`mode`is unspecified return the Cholesky factorization for`R`. If`mode`is`"lower"`

or`"L"`

then return the conjugate transpose

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:`R`'`count`,`h`,`parent`, and`post`.

The output variables are:

`count`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)`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`.

**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.

- :
`s`=**spaugment***(*¶`A`,`c`) 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

, by`A`\`b``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 (`

for underdetermined problems, and example can be`m`,`m`)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.

- :
`d`=**eigs***(*¶`A`) - :
`d`=**eigs***(*¶`A`,`k`) - :
`d`=**eigs***(*¶`A`,`k`,`sigma`) - :
`d`=**eigs***(*¶`A`,`k`,`sigma`,`opts`) - :
`d`=**eigs***(*¶`A`,`B`) - :
`d`=**eigs***(*¶`A`,`B`,`k`) - :
`d`=**eigs***(*¶`A`,`B`,`k`,`sigma`) - :
`d`=**eigs***(*¶`A`,`B`,`k`,`sigma`,`opts`) - :
`d`=**eigs***(*¶`Af`,`n`) - :
`d`=**eigs***(*¶`Af`,`n`,`k`) - :
`d`=**eigs***(*¶`Af`,`n`,`k`,`sigma`) - :
`d`=**eigs***(*¶`Af`,`n`,`k`,`sigma`,`opts`) - :
`d`=**eigs***(*¶`Af`,`n`,`B`) - :
`d`=**eigs***(*¶`Af`,`n`,`B`,`k`) - :
`d`=**eigs***(*¶`Af`,`n`,`B`,`k`,`sigma`) - :
`d`=**eigs***(*¶`Af`,`n`,`B`,`k`,`sigma`,`opts`) - :
*[*`V`,`D`] =**eigs***(…)*¶ - :
*[*`V`,`D`,`flag`] =**eigs***(…)*¶ Calculate a limited number of eigenvalues and eigenvectors based on a selection criteria.

By default,

`eigs`

solve the equation where is the corresponding eigenvector. If given the positive definite matrix`B`then`eigs`

solves the general eigenvalue equationThe input

`A`is a square matrix of dimension`n`-by-`n`. Typically,`A`is also large and sparse.The input

`B`for the generalized eigenvalue problem is a square matrix with the same size as`A`(`n`-by-`n`). Typically,`B`is also large and sparse.The number of eigenvalues and eigenvectors to calculate is given by

`k`and defaults to 6.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 be 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 this flag (true/false) determines whether the function`Af`defines a symmetric problem. It is ignored if a matrix`A`is given. The default is false.`isreal`

If

`Af`is given then this flag (true/false) determines whether the function`Af`defines a real problem. It is ignored if a matrix`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 Lanczos 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

to`k`+ 1`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`

If the generalized eigenvalue problem is being calculated, this flag (true/false) specifies whether the

`B`input represents`chol (`

or simply the matrix`B`)`B`. The default is false.`permB`

The permutation vector of the Cholesky factorization for

`B`if`cholB`

is true. It is obtained by`[R, ~, permB] = chol (`

. The default is`B`,`"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`

if

`sigma`is a scalar not equal to 0;`I`

is the identity matrix of the same size as`A`.`(A - sigma * B) \ x`

for the general eigenvalue problem.

The return arguments and their form depend on the number of return arguments requested. For a single return argument, a column vector

`d`of length`k`is returned containing the`k`eigenvalues that have been found. For two return arguments,`V`is an`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`k`-by-`k`matrix, where the elements on the diagonal are the eigenvalues.The third return argument

`flag`returns the status of the convergence. If`flag`is 0 then all eigenvalues have converged. Any other value indicates a failure to converge.Programming Notes: For small problems,

`n`< 500, consider using`eig (full (`

.`A`))If ARPACK fails to converge consider increasing the number of Lanczos vectors (

`opt`.p), increasing the number of iterations (`opt`.maxiter), or decreasing the tolerance (`opt`.tol).Reference: 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/.

- :
`s`=**svds***(*¶`A`) - :
`s`=**svds***(*¶`A`,`k`) - :
`s`=**svds***(*¶`A`,`k`,`sigma`) - :
`s`=**svds***(*¶`A`,`k`,`sigma`,`opts`) - :
*[*`u`,`s`,`v`] =**svds***(…)*¶ - :
*[*`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`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,

finds the smallest singular values. Note that for relatively small values of`sigma`= 0`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 isnorm (

`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 (`

will likely be more efficient.`A`))

The CHOLMOD, UMFPACK and CXSPARSE packages were written by Tim Davis and are available at http://faculty.cse.tamu.edu/davis/suitesparse.html