Class SymmLQ
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 selfadjoint 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 Rayleighquotient iteration. Again, the linear operator (A  shift · I) need not be positive definite (but must be selfadjoint). 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 = P^{T} · P that is known to approximate (A  shift · I)^{1} in some sense, where matrixvector products of the form M · y = x can be computed efficiently. Then SYMMLQ will implicitly solve the system of equations P · (A  shift · I) · P^{T} · x_{hat} = P · b, i.e. A_{hat} · x_{hat} = b_{hat}, where A_{hat} = P · (A  shift · I) · P^{T}, b_{hat} = P · b, and return the solution x = P^{T} · x_{hat}. The associated residual is r_{hat} = b_{hat}  A_{hat} · x_{hat} = P · [b  (A  shift · I) · x] = P · r.
In the case of preconditioning, the IterativeLinearSolverEvent
s 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 userspecified tolerance.
Iteration countIn the present context, an iteration should be understood as one evaluation of the matrixvector 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 matrixvector 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
solve(RealLinearOperator, RealVector, RealVector)
,solve(RealLinearOperator, RealLinearOperator, RealVector, RealVector)
},solveInPlace(RealLinearOperator, RealVector, RealVector)
,solveInPlace(RealLinearOperator, RealLinearOperator, RealVector, RealVector)
,solveInPlace(RealLinearOperator, RealLinearOperator, RealVector, RealVector, boolean, double)
,
should not be considered as an initial guess, as it is set to zero in the initial phase. If x_{0} is known to be a good approximation to x, one should compute r_{0} = b  A · x, solve A · dx = r0, and set x = x_{0} + dx.
Exception context
Besides standard MathIllegalArgumentException
, this class might throw
MathIllegalArgumentException
if the linear operator or the
preconditioner are not symmetric.
 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 x^{T} · L · y ≠ y^{T} · L · x (within a certain accuracy).
MathIllegalArgumentException
might also be thrown in case the
preconditioner is not positive definite.
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): 617629, 1975

Constructor Summary
ConstructorDescriptionSymmLQ
(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
Modifier and TypeMethodDescriptionfinal boolean
Returnstrue
if symmetry of the matrix, and symmetry as well as positive definiteness of the preconditioner should be checked.Returns an estimate of the solution to the linear system A · x = b.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.solve
(RealLinearOperator a, RealLinearOperator m, RealVector b, RealVector x) Returns an estimate of the solution to the linear system A · x = b.Returns an estimate of the solution to the linear system A · x = b.solve
(RealLinearOperator a, RealVector b, boolean goodb, double shift) Returns the solution to the system (A  shift · I) · x = b.solve
(RealLinearOperator a, RealVector b, RealVector x) Returns an estimate of the solution to the linear system A · x = b.Returns an estimate of the solution to the linear system A · x = b.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.Returns an estimate of the solution to the linear system A · x = b.Methods inherited from class org.hipparchus.linear.PreconditionedIterativeLinearSolver
checkParameters
Methods inherited from class org.hipparchus.linear.IterativeLinearSolver
checkParameters, getIterationManager

Constructor Details

SymmLQ
public SymmLQ(int maxIterations, double delta, boolean check) Creates a new instance of this class, with default stopping criterion. Note that settingcheck
totrue
entails an extra matrixvector product in the initial phase. Parameters:
maxIterations
 the maximum number of iterationsdelta
 the δ parameter for the default stopping criterioncheck
true
if selfadjointedness of both matrix and preconditioner should be checked

SymmLQ
Creates a new instance of this class, with default stopping criterion and custom iteration manager. Note that settingcheck
totrue
entails an extra matrixvector product in the initial phase. Parameters:
manager
 the custom iteration managerdelta
 the δ parameter for the default stopping criterioncheck
true
if selfadjointedness of both matrix and preconditioner should be checked


Method Details

shouldCheck
public final boolean shouldCheck()Returnstrue
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 Since:
 1.4

solve
public RealVector solve(RealLinearOperator a, RealLinearOperator m, RealVector b) throws MathIllegalArgumentException, NullArgumentException, MathIllegalStateException, MathIllegalArgumentException Returns an estimate of the solution to the linear system A · x = b. Overrides:
solve
in classPreconditionedIterativeLinearSolver
 Parameters:
a
 the linear operator A of the systemm
 the preconditioner, M (can benull
)b
 the righthand side vector Returns:
 a new vector containing the solution
 Throws:
MathIllegalArgumentException
 ifshouldCheck()
istrue
, anda
orm
is not selfadjointMathIllegalArgumentException
 ifm
is not positive definiteMathIllegalArgumentException
 ifa
is illconditionedNullArgumentException
 if one of the parameters isnull
MathIllegalStateException
 at exhaustion of the iteration count, unless a customcallback
has been set at construction of theIterationManager

solve
public RealVector solve(RealLinearOperator a, RealLinearOperator m, RealVector b, boolean goodb, double shift) throws MathIllegalArgumentException, NullArgumentException, MathIllegalStateException 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 Rayleighquotient iteration), then better precision may be achieved withgoodb
set totrue
; 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 b^{T} · A · b / (b^{T} · 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 systemm
 the preconditioner, M (can benull
)b
 the righthand side vectorgoodb
 usuallyfalse
, except ifx
is expected to contain a large multiple ofb
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 isnull
MathIllegalArgumentException
 ifa
orm
is not squareMathIllegalArgumentException
 ifm
orb
have dimensions inconsistent witha
MathIllegalStateException
 at exhaustion of the iteration count, unless a customcallback
has been set at construction of theIterationManager
MathIllegalArgumentException
 ifshouldCheck()
istrue
, anda
orm
is not selfadjointMathIllegalArgumentException
 ifm
is not positive definiteMathIllegalArgumentException
 ifa
is illconditioned

solve
public RealVector solve(RealLinearOperator a, RealLinearOperator m, RealVector b, RealVector x) throws MathIllegalArgumentException, NullArgumentException, MathIllegalArgumentException, MathIllegalStateException Returns an estimate of the solution to the linear system A · x = b. Overrides:
solve
in classPreconditionedIterativeLinearSolver
 Parameters:
x
 not meaningful in this implementation; should not be considered as an initial guess (more)a
 the linear operator A of the systemm
 the preconditioner, M (can benull
)b
 the righthand side vector Returns:
 a new vector containing the solution
 Throws:
MathIllegalArgumentException
 ifshouldCheck()
istrue
, anda
orm
is not selfadjointMathIllegalArgumentException
 ifm
is not positive definiteMathIllegalArgumentException
 ifa
is illconditionedNullArgumentException
 if one of the parameters isnull
MathIllegalStateException
 at exhaustion of the iteration count, unless a customcallback
has been set at construction of theIterationManager

solve
public RealVector solve(RealLinearOperator a, RealVector b) throws MathIllegalArgumentException, NullArgumentException, MathIllegalArgumentException, MathIllegalStateException Returns an estimate of the solution to the linear system A · x = b. Overrides:
solve
in classPreconditionedIterativeLinearSolver
 Parameters:
a
 the linear operator A of the systemb
 the righthand side vector Returns:
 a new vector containing the solution
 Throws:
MathIllegalArgumentException
 ifshouldCheck()
istrue
, anda
is not selfadjointMathIllegalArgumentException
 ifa
is illconditionedNullArgumentException
 if one of the parameters isnull
MathIllegalStateException
 at exhaustion of the iteration count, unless a customcallback
has been set at construction of theIterationManager

solve
public RealVector solve(RealLinearOperator a, RealVector b, boolean goodb, double shift) throws MathIllegalArgumentException, NullArgumentException, MathIllegalStateException 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 Rayleighquotient iteration), then better precision may be achieved withgoodb
set totrue
.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 b^{T} · A · b / (b^{T} · 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 systemb
 the righthand side vectorgoodb
 usuallyfalse
, except ifx
is expected to contain a large multiple ofb
shift
 the amount to be subtracted to all diagonal elements of A Returns:
 a reference to
x
 Throws:
NullArgumentException
 if one of the parameters isnull
MathIllegalArgumentException
 ifa
is not squareMathIllegalArgumentException
 ifb
has dimensions inconsistent witha
MathIllegalStateException
 at exhaustion of the iteration count, unless a customcallback
has been set at construction of theIterationManager
MathIllegalArgumentException
 ifshouldCheck()
istrue
, anda
is not selfadjointMathIllegalArgumentException
 ifa
is illconditioned

solve
public RealVector solve(RealLinearOperator a, RealVector b, RealVector x) throws MathIllegalArgumentException, NullArgumentException, MathIllegalArgumentException, MathIllegalStateException Returns an estimate of the solution to the linear system A · x = b. Overrides:
solve
in classPreconditionedIterativeLinearSolver
 Parameters:
x
 not meaningful in this implementation; should not be considered as an initial guess (more)a
 the linear operator A of the systemb
 the righthand side vector Returns:
 a new vector containing the solution
 Throws:
MathIllegalArgumentException
 ifshouldCheck()
istrue
, anda
is not selfadjointMathIllegalArgumentException
 ifa
is illconditionedNullArgumentException
 if one of the parameters isnull
MathIllegalStateException
 at exhaustion of the iteration count, unless a customcallback
has been set at construction of theIterationManager

solveInPlace
public RealVector solveInPlace(RealLinearOperator a, RealLinearOperator m, RealVector b, RealVector x) throws MathIllegalArgumentException, NullArgumentException, MathIllegalArgumentException, MathIllegalStateException Returns an estimate of the solution to the linear system A · x = b. The solution is computed inplace (initial guess is modified). Specified by:
solveInPlace
in classPreconditionedIterativeLinearSolver
 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 systemm
 the preconditioner, M (can benull
)b
 the righthand side vector Returns:
 a reference to
x0
(shallow copy) updated with the solution  Throws:
MathIllegalArgumentException
 ifshouldCheck()
istrue
, anda
orm
is not selfadjointMathIllegalArgumentException
 ifm
is not positive definiteMathIllegalArgumentException
 ifa
is illconditionedNullArgumentException
 if one of the parameters isnull
MathIllegalStateException
 at exhaustion of the iteration count, unless a customcallback
has been set at construction of theIterationManager

solveInPlace
public RealVector solveInPlace(RealLinearOperator a, RealLinearOperator m, RealVector b, RealVector x, boolean goodb, double shift) throws MathIllegalArgumentException, NullArgumentException, MathIllegalStateException Returns an estimate of the solution to the linear system (A  shift · I) · x = b. The solution is computed inplace.If the solution x is expected to contain a large multiple of
b
(as in Rayleighquotient iteration), then better precision may be achieved withgoodb
set totrue
; 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 b^{T} · A · b / (b^{T} · 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 systemm
 the preconditioner, M (can benull
)b
 the righthand side vectorx
 the vector to be updated with the solution;x
should not be considered as an initial guess (more)goodb
 usuallyfalse
, except ifx
is expected to contain a large multiple ofb
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 isnull
MathIllegalArgumentException
 ifa
orm
is not squareMathIllegalArgumentException
 ifm
,b
orx
have dimensions inconsistent witha
.MathIllegalStateException
 at exhaustion of the iteration count, unless a customcallback
has been set at construction of theIterationManager
MathIllegalArgumentException
 ifshouldCheck()
istrue
, anda
orm
is not selfadjointMathIllegalArgumentException
 ifm
is not positive definiteMathIllegalArgumentException
 ifa
is illconditioned

solveInPlace
public RealVector solveInPlace(RealLinearOperator a, RealVector b, RealVector x) throws MathIllegalArgumentException, NullArgumentException, MathIllegalArgumentException, MathIllegalStateException Returns an estimate of the solution to the linear system A · x = b. The solution is computed inplace (initial guess is modified). Overrides:
solveInPlace
in classPreconditionedIterativeLinearSolver
 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 systemb
 the righthand side vector Returns:
 a reference to
x0
(shallow copy) updated with the solution  Throws:
MathIllegalArgumentException
 ifshouldCheck()
istrue
, anda
is not selfadjointMathIllegalArgumentException
 ifa
is illconditionedNullArgumentException
 if one of the parameters isnull
MathIllegalStateException
 at exhaustion of the iteration count, unless a customcallback
has been set at construction of theIterationManager
