org.apache.commons.math3.linear

## Class SymmLQ

• public class SymmLQ
extends PreconditionedIterativeLinearSolver

Implementation of the SYMMLQ iterative linear solver proposed by Paige and Saunders (1975). This implementation is largely based on the FORTRAN code by Pr. Michael A. Saunders, available here.

SYMMLQ is designed to solve the system of linear equations A · x = b where A is an n × n self-adjoint linear operator (defined as a RealLinearOperator), and b is a given vector. The operator A is not required to be positive definite. If A is known to be definite, the method of conjugate gradients might be preferred, since it will require about the same number of iterations as SYMMLQ but slightly less work per iteration.

SYMMLQ is designed to solve the system (A - shift · I) · x = b, where shift is a specified scalar value. If shift and b are suitably chosen, the computed vector x may approximate an (unnormalized) eigenvector of A, as in the methods of inverse iteration and/or Rayleigh-quotient iteration. Again, the linear operator (A - shift · I) need not be positive definite (but must be self-adjoint). The work per iteration is very slightly less if shift = 0.

### Preconditioning

Preconditioning may reduce the number of iterations required. The solver may be provided with a positive definite preconditioner M = PT · P that is known to approximate (A - shift · I)-1 in some sense, where matrix-vector products of the form M · y = x can be computed efficiently. Then SYMMLQ will implicitly solve the system of equations P · (A - shift · I) · PT · xhat = P · b, i.e. Ahat · xhat = bhat, where Ahat = P · (A - shift · I) · PT, bhat = P · b, and return the solution x = PT · xhat. The associated residual is rhat = bhat - Ahat · xhat = P · [b - (A - shift · I) · x] = P · r.

In the case of preconditioning, the IterativeLinearSolverEvents that this solver fires are such that IterativeLinearSolverEvent.getNormOfResidual() returns the norm of the preconditioned, updated residual, ||P · r||, not the norm of the true residual ||r||.

### Default stopping criterion

A default stopping criterion is implemented. The iterations stop when || rhat || ≤ δ || Ahat || || xhat ||, where xhat is the current estimate of the solution of the transformed system, rhat the current estimate of the corresponding residual, and δ a user-specified tolerance.

### Iteration count

In the present context, an iteration should be understood as one evaluation of the matrix-vector product A · x. The initialization phase therefore counts as one iteration. If the user requires checks on the symmetry of A, this entails one further matrix-vector product in the initial phase. This further product is not accounted for in the iteration count. In other words, the number of iterations required to reach convergence will be identical, whether checks have been required or not.

The present definition of the iteration count differs from that adopted in the original FOTRAN code, where the initialization phase was not taken into account.

### Initial guess of the solution

The x parameter in

should not be considered as an initial guess, as it is set to zero in the initial phase. If x0 is known to be a good approximation to x, one should compute r0 = b - A · x, solve A · dx = r0, and set x = x0 + dx.

### Exception context

Besides standard DimensionMismatchException, this class might throw NonSelfAdjointOperatorException if the linear operator or the preconditioner are not symmetric. In this case, the ExceptionContext provides more information

• key "operator" points to the offending linear operator, say L,
• key "vector1" points to the first offending vector, say x,
• key "vector2" points to the second offending vector, say y, such that xT · L · y ≠ yT · L · x (within a certain accuracy).

NonPositiveDefiniteOperatorException might also be thrown in case the preconditioner is not positive definite. The relevant keys to the ExceptionContext are

• key "operator", which points to the offending linear operator, say L,
• key "vector", which points to the offending vector, say x, such that xT · L · x < 0.

### References

Paige and Saunders (1975)
C. C. Paige and M. A. Saunders, Solution of Sparse Indefinite Systems of Linear Equations, SIAM Journal on Numerical Analysis 12(4): 617-629, 1975
Since:
3.0
Version:
$Id: SymmLQ.java 1505938 2013-07-23 08:50:10Z luc$
• ### Constructor Summary

Constructors
Constructor and Description
SymmLQ(int maxIterations, double delta, boolean check)
Creates a new instance of this class, with default stopping criterion.
SymmLQ(IterationManager manager, double delta, boolean check)
Creates a new instance of this class, with default stopping criterion and custom iteration manager.
• ### Method Summary

Methods
Modifier and Type Method and Description
boolean getCheck()
Returns true if symmetry of the matrix, and symmetry as well as positive definiteness of the preconditioner should be checked.
RealVector solve(RealLinearOperator a, RealLinearOperator m, RealVector b)
Returns an estimate of the solution to the linear system A · x = b.
RealVector solve(RealLinearOperator a, RealLinearOperator m, RealVector b, boolean goodb, double shift)
Returns an estimate of the solution to the linear system (A - shift · I) · x = b.
RealVector solve(RealLinearOperator a, RealLinearOperator m, RealVector b, RealVector x)
Returns an estimate of the solution to the linear system A · x = b.
RealVector solve(RealLinearOperator a, RealVector b)
Returns an estimate of the solution to the linear system A · x = b.
RealVector solve(RealLinearOperator a, RealVector b, boolean goodb, double shift)
Returns the solution to the system (A - shift · I) · x = b.
RealVector solve(RealLinearOperator a, RealVector b, RealVector x)
Returns an estimate of the solution to the linear system A · x = b.
RealVector solveInPlace(RealLinearOperator a, RealLinearOperator m, RealVector b, RealVector x)
Returns an estimate of the solution to the linear system A · x = b.
RealVector solveInPlace(RealLinearOperator a, RealLinearOperator m, RealVector b, RealVector x, boolean goodb, double shift)
Returns an estimate of the solution to the linear system (A - shift · I) · x = b.
RealVector solveInPlace(RealLinearOperator a, RealVector b, RealVector x)
Returns an estimate of the solution to the linear system A · x = b.
• ### Methods inherited from class org.apache.commons.math3.linear.PreconditionedIterativeLinearSolver

checkParameters
• ### Methods inherited from class org.apache.commons.math3.linear.IterativeLinearSolver

checkParameters, getIterationManager
• ### Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
• ### Constructor Detail

• #### SymmLQ

public SymmLQ(int maxIterations,
double delta,
boolean check)
Creates a new instance of this class, with default stopping criterion. Note that setting check to true entails an extra matrix-vector product in the initial phase.
Parameters:
maxIterations - the maximum number of iterations
delta - the δ parameter for the default stopping criterion
check - true if self-adjointedness of both matrix and preconditioner should be checked
• #### SymmLQ

public SymmLQ(IterationManager manager,
double delta,
boolean check)
Creates a new instance of this class, with default stopping criterion and custom iteration manager. Note that setting check to true entails an extra matrix-vector product in the initial phase.
Parameters:
manager - the custom iteration manager
delta - the δ parameter for the default stopping criterion
check - true if self-adjointedness of both matrix and preconditioner should be checked
• ### Method Detail

• #### getCheck

public final boolean getCheck()
Returns true if symmetry of the matrix, and symmetry as well as positive definiteness of the preconditioner should be checked.
Returns:
true if the tests are to be performed
• #### solve

public RealVector solve(RealLinearOperator a,
RealLinearOperator m,
RealVector b)
throws NullArgumentException,
NonSquareOperatorException,
DimensionMismatchException,
MaxCountExceededException,
NonPositiveDefiniteOperatorException,
IllConditionedOperatorException
Returns an estimate of the solution to the linear system A · x = b.
Overrides:
solve in class PreconditionedIterativeLinearSolver
Parameters:
a - the linear operator A of the system
m - the preconditioner, M (can be null)
b - the right-hand side vector
Returns:
a new vector containing the solution
Throws:
NonSelfAdjointOperatorException - if getCheck() is true, and a or m is not self-adjoint
NonPositiveDefiniteOperatorException - if m is not positive definite
IllConditionedOperatorException - if a is ill-conditioned
NullArgumentException - if one of the parameters is null
NonSquareOperatorException - if a or m is not square
DimensionMismatchException - if m or b have dimensions inconsistent with a
MaxCountExceededException - at exhaustion of the iteration count, unless a custom callback has been set at construction of the IterationManager
• #### solve

public RealVector solve(RealLinearOperator a,
RealLinearOperator m,
RealVector b,
boolean goodb,
double shift)
throws NullArgumentException,
NonSquareOperatorException,
DimensionMismatchException,
MaxCountExceededException,
NonPositiveDefiniteOperatorException,
IllConditionedOperatorException
Returns an estimate of the solution to the linear system (A - shift · I) · x = b.

If the solution x is expected to contain a large multiple of b (as in Rayleigh-quotient iteration), then better precision may be achieved with goodb set to true; this however requires an extra call to the preconditioner.

shift should be zero if the system A · x = b is to be solved. Otherwise, it could be an approximation to an eigenvalue of A, such as the Rayleigh quotient bT · A · b / (bT · b) corresponding to the vector b. If b is sufficiently like an eigenvector corresponding to an eigenvalue near shift, then the computed x may have very large components. When normalized, x may be closer to an eigenvector than b.

Parameters:
a - the linear operator A of the system
m - the preconditioner, M (can be null)
b - the right-hand side vector
goodb - usually false, except if x is expected to contain a large multiple of b
shift - the amount to be subtracted to all diagonal elements of A
Returns:
a reference to x (shallow copy)
Throws:
NullArgumentException - if one of the parameters is null
NonSquareOperatorException - if a or m is not square
DimensionMismatchException - if m or b have dimensions inconsistent with a
MaxCountExceededException - at exhaustion of the iteration count, unless a custom callback has been set at construction of the IterationManager
NonSelfAdjointOperatorException - if getCheck() is true, and a or m is not self-adjoint
NonPositiveDefiniteOperatorException - if m is not positive definite
IllConditionedOperatorException - if a is ill-conditioned
• #### solve

public RealVector solve(RealLinearOperator a,
RealLinearOperator m,
RealVector b,
RealVector x)
throws NullArgumentException,
NonSquareOperatorException,
DimensionMismatchException,
NonPositiveDefiniteOperatorException,
IllConditionedOperatorException,
MaxCountExceededException
Returns an estimate of the solution to the linear system A · x = b.
Overrides:
solve in class PreconditionedIterativeLinearSolver
Parameters:
x - not meaningful in this implementation; should not be considered as an initial guess (more)
a - the linear operator A of the system
m - the preconditioner, M (can be null)
b - the right-hand side vector
Returns:
a new vector containing the solution
Throws:
NonSelfAdjointOperatorException - if getCheck() is true, and a or m is not self-adjoint
NonPositiveDefiniteOperatorException - if m is not positive definite
IllConditionedOperatorException - if a is ill-conditioned
NullArgumentException - if one of the parameters is null
NonSquareOperatorException - if a or m is not square
DimensionMismatchException - if m, b or x0 have dimensions inconsistent with a
MaxCountExceededException - at exhaustion of the iteration count, unless a custom callback has been set at construction of the IterationManager
• #### solve

public RealVector solve(RealLinearOperator a,
RealVector b)
throws NullArgumentException,
NonSquareOperatorException,
DimensionMismatchException,
IllConditionedOperatorException,
MaxCountExceededException
Returns an estimate of the solution to the linear system A · x = b.
Overrides:
solve in class PreconditionedIterativeLinearSolver
Parameters:
a - the linear operator A of the system
b - the right-hand side vector
Returns:
a new vector containing the solution
Throws:
NonSelfAdjointOperatorException - if getCheck() is true, and a is not self-adjoint
IllConditionedOperatorException - if a is ill-conditioned
NullArgumentException - if one of the parameters is null
NonSquareOperatorException - if a is not square
DimensionMismatchException - if b has dimensions inconsistent with a
MaxCountExceededException - at exhaustion of the iteration count, unless a custom callback has been set at construction of the IterationManager
• #### solve

public RealVector solve(RealLinearOperator a,
RealVector b,
boolean goodb,
double shift)
throws NullArgumentException,
NonSquareOperatorException,
DimensionMismatchException,
IllConditionedOperatorException,
MaxCountExceededException
Returns the solution to the system (A - shift · I) · x = b.

If the solution x is expected to contain a large multiple of b (as in Rayleigh-quotient iteration), then better precision may be achieved with goodb set to true.

shift should be zero if the system A · x = b is to be solved. Otherwise, it could be an approximation to an eigenvalue of A, such as the Rayleigh quotient bT · A · b / (bT · b) corresponding to the vector b. If b is sufficiently like an eigenvector corresponding to an eigenvalue near shift, then the computed x may have very large components. When normalized, x may be closer to an eigenvector than b.

Parameters:
a - the linear operator A of the system
b - the right-hand side vector
goodb - usually false, except if x is expected to contain a large multiple of b
shift - the amount to be subtracted to all diagonal elements of A
Returns:
a reference to x
Throws:
NullArgumentException - if one of the parameters is null
NonSquareOperatorException - if a is not square
DimensionMismatchException - if b has dimensions inconsistent with a
MaxCountExceededException - at exhaustion of the iteration count, unless a custom callback has been set at construction of the IterationManager
NonSelfAdjointOperatorException - if getCheck() is true, and a is not self-adjoint
IllConditionedOperatorException - if a is ill-conditioned
• #### solve

public RealVector solve(RealLinearOperator a,
RealVector b,
RealVector x)
throws NullArgumentException,
NonSquareOperatorException,
DimensionMismatchException,
IllConditionedOperatorException,
MaxCountExceededException
Returns an estimate of the solution to the linear system A · x = b.
Overrides:
solve in class PreconditionedIterativeLinearSolver
Parameters:
x - not meaningful in this implementation; should not be considered as an initial guess (more)
a - the linear operator A of the system
b - the right-hand side vector
Returns:
a new vector containing the solution
Throws:
NonSelfAdjointOperatorException - if getCheck() is true, and a is not self-adjoint
IllConditionedOperatorException - if a is ill-conditioned
NullArgumentException - if one of the parameters is null
NonSquareOperatorException - if a is not square
DimensionMismatchException - if b or x0 have dimensions inconsistent with a
MaxCountExceededException - at exhaustion of the iteration count, unless a custom callback has been set at construction of the IterationManager
• #### solveInPlace

public RealVector solveInPlace(RealLinearOperator a,
RealLinearOperator m,
RealVector b,
RealVector x)
throws NullArgumentException,
NonSquareOperatorException,
DimensionMismatchException,
NonPositiveDefiniteOperatorException,
IllConditionedOperatorException,
MaxCountExceededException
Returns an estimate of the solution to the linear system A · x = b. The solution is computed in-place (initial guess is modified).
Specified by:
solveInPlace in class PreconditionedIterativeLinearSolver
Parameters:
x - the vector to be updated with the solution; x should not be considered as an initial guess (more)
a - the linear operator A of the system
m - the preconditioner, M (can be null)
b - the right-hand side vector
Returns:
a reference to x0 (shallow copy) updated with the solution
Throws:
NonSelfAdjointOperatorException - if getCheck() is true, and a or m is not self-adjoint
NonPositiveDefiniteOperatorException - if m is not positive definite
IllConditionedOperatorException - if a is ill-conditioned
NullArgumentException - if one of the parameters is null
NonSquareOperatorException - if a or m is not square
DimensionMismatchException - if m, b or x0 have dimensions inconsistent with a
MaxCountExceededException - at exhaustion of the iteration count, unless a custom callback has been set at construction of the IterationManager
• #### solveInPlace

public RealVector solveInPlace(RealLinearOperator a,
RealLinearOperator m,
RealVector b,
RealVector x,
boolean goodb,
double shift)
throws NullArgumentException,
NonSquareOperatorException,
DimensionMismatchException,
NonPositiveDefiniteOperatorException,
IllConditionedOperatorException,
MaxCountExceededException
Returns an estimate of the solution to the linear system (A - shift · I) · x = b. The solution is computed in-place.

If the solution x is expected to contain a large multiple of b (as in Rayleigh-quotient iteration), then better precision may be achieved with goodb set to true; this however requires an extra call to the preconditioner.

shift should be zero if the system A · x = b is to be solved. Otherwise, it could be an approximation to an eigenvalue of A, such as the Rayleigh quotient bT · A · b / (bT · b) corresponding to the vector b. If b is sufficiently like an eigenvector corresponding to an eigenvalue near shift, then the computed x may have very large components. When normalized, x may be closer to an eigenvector than b.

Parameters:
a - the linear operator A of the system
m - the preconditioner, M (can be null)
b - the right-hand side vector
x - the vector to be updated with the solution; x should not be considered as an initial guess (more)
goodb - usually false, except if x is expected to contain a large multiple of b
shift - the amount to be subtracted to all diagonal elements of A
Returns:
a reference to x (shallow copy).
Throws:
NullArgumentException - if one of the parameters is null
NonSquareOperatorException - if a or m is not square
DimensionMismatchException - if m, b or x have dimensions inconsistent with a.
MaxCountExceededException - at exhaustion of the iteration count, unless a custom callback has been set at construction of the IterationManager
NonSelfAdjointOperatorException - if getCheck() is true, and a or m is not self-adjoint
NonPositiveDefiniteOperatorException - if m is not positive definite
IllConditionedOperatorException - if a is ill-conditioned
• #### solveInPlace

public RealVector solveInPlace(RealLinearOperator a,
RealVector b,
RealVector x)
throws NullArgumentException,
NonSquareOperatorException,
DimensionMismatchException,
IllConditionedOperatorException,
MaxCountExceededException
Returns an estimate of the solution to the linear system A · x = b. The solution is computed in-place (initial guess is modified).
Overrides:
solveInPlace in class PreconditionedIterativeLinearSolver
Parameters:
x - the vector to be updated with the solution; x should not be considered as an initial guess (more)
a - the linear operator A of the system
b - the right-hand side vector
Returns:
a reference to x0 (shallow copy) updated with the solution
Throws:
NonSelfAdjointOperatorException - if getCheck() is true, and a is not self-adjoint
IllConditionedOperatorException - if a is ill-conditioned
NullArgumentException - if one of the parameters is null
NonSquareOperatorException - if a is not square
DimensionMismatchException - if b or x0 have dimensions inconsistent with a
MaxCountExceededException - at exhaustion of the iteration count, unless a custom callback has been set at construction of the IterationManager