com.imsl.math

Class OdeAdamsGear

java.lang.Object

com.imsl.math.ODE

com.imsl.math.OdeAdamsGear

All Implemented Interfaces:

Serializable, Cloneable

public class OdeAdamsGear
extends ODE

Extension of the ODE class to solve a stiff initial-value problem for ordinary differential equations using the Adams-Gear methods.

Class OdeAdamsGear finds an approximation to the solution of a system of first-order differential equations of the form

$$ \frac{dy}{dt} = y' = f(t,y) $$

with given initial conditions for \(y\) at the starting value for \(t\). The class attempts to keep the global error proportional to a user-specified tolerance. The proportionality depends on the differential equation and the range of integration.

The code is based on using backward difference formulas not exceeding order five as outlined in Gear (1971) and implemented by Hindmarsh (1974). There is an optional use of the code that employs implicit Adams formulas. This use is intended for nonstiff problems with expensive functions \( y' = f(t,y) \).

See Also:

Example, Serialized Form

Nested Class Summary

Nested Classes

Modifier and Type	Class and Description
static class	OdeAdamsGear.DidNotConvergeException The iteration did not converge within the maximum number of steps allowed (default 500).
static interface	OdeAdamsGear.Function Public interface for user supplied function to OdeAdamsGear object.
static interface	OdeAdamsGear.Jacobian Public interface for the user supplied function to evaluate the Jacobian matrix.
static class	OdeAdamsGear.MaxFcnEvalsExceededException Maximum function evaluations exceeded.
static class	OdeAdamsGear.SingularMatrixException The interpolation matrix is singular.
static class	OdeAdamsGear.ToleranceTooSmallException Tolerance is too small or the problem is stiff.

Field Summary

Fields

Modifier and Type	Field and Description
static int	METHOD_ADAMS The Adams integration method
static int	METHOD_BDF The BDF integration method
static int	SOLVE_CHORD_COMPUTED_DIAGONAL A chord method and a diagonal matrix based on a directional directive
static int	SOLVE_CHORD_COMPUTED_JACOBIAN A chord or modified Newton method and a divided differences Jacobian
static int	SOLVE_CHORD_USER_JACOBIAN A chord or modified Newton method and a user-supplied Jacobian
static int	SOLVE_FUNCTION_ITERATION A function iteration or successive substitution method

Fields inherited from class com.imsl.math.ODE

AFTER_SUCCESSFUL_STEP, AFTER_UNSUCCESSFUL_STEP, BEFORE_STEP, ERROR_NORM_ABS, ERROR_NORM_EUCLIDEAN, ERROR_NORM_MAX, ERROR_NORM_MINABSREL

Constructor Summary

Constructors

Constructor and Description
OdeAdamsGear(OdeAdamsGear.Function function) Constructs an ODE solver to solve the initial value problem dy/dt = f(t,y)

Method Summary

Modifier and Type	Method and Description
int	getIntegrationMethod() Returns the integration method used.
int	getMaximumFunctionEvaluations() Returns the maximum number of function evaluations of \(y'\) allowed.
int	getMaxOrder() Returns the highest order formula to use of implicit METHOD_ADAMS type or METHOD_BDF type.
int	getNumberOfFcnEvals() Returns the number of function evaluations of \(y'\) made.
int	getNumberOfJacobianEvals() Returns the number of Jacobian matrix evaluations used.
int	getNumberOfSteps() Returns the number of internal steps taken.
int	getSolveMethod() Returns the method for solving the formula equations.
void	setIntegrationMethod(int intMethod) Indicates which integration method is to be used.
void	setMaximumFunctionEvaluations(int maxfcn) Sets the maximum number of function evaluations of \(y'\) allowed.
void	setMaximumStepsize(double stepsize) Sets the maximum internal step size.
void	setMaxOrder(int maxOrder) Sets the highest order formula to use of implicit METHOD_ADAMS type or METHOD_BDF type.
void	setSolveMethod(int solveMethod) Indicates which method to use for solving the formula equations.
void	solve(double t, double tEnd, double[] y) Integrates the ODE system from t to tEnd.

Methods inherited from class com.imsl.math.ODE

examineStep, getFloor, getInitialStepsize, getMaximumStepsize, getMaxSteps, getMinimumStepsize, getNorm, getScale, getTolerance, setFloor, setInitialStepsize, setMaxSteps, setMinimumStepsize, setNorm, setScale, setTolerance, vnorm

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Field Detail

METHOD_ADAMS

public static final int METHOD_ADAMS

The Adams integration method

See Also:

Constant Field Values

METHOD_BDF

public static final int METHOD_BDF

The BDF integration method

See Also:

Constant Field Values

SOLVE_FUNCTION_ITERATION

public static final int SOLVE_FUNCTION_ITERATION

A function iteration or successive substitution method

See Also:

Constant Field Values

SOLVE_CHORD_USER_JACOBIAN

public static final int SOLVE_CHORD_USER_JACOBIAN

A chord or modified Newton method and a user-supplied Jacobian

See Also:

Constant Field Values

SOLVE_CHORD_COMPUTED_JACOBIAN

public static final int SOLVE_CHORD_COMPUTED_JACOBIAN

A chord or modified Newton method and a divided differences Jacobian

See Also:

Constant Field Values

SOLVE_CHORD_COMPUTED_DIAGONAL

public static final int SOLVE_CHORD_COMPUTED_DIAGONAL

A chord method and a diagonal matrix based on a directional directive

See Also:

Constant Field Values

Constructor Detail

OdeAdamsGear

public OdeAdamsGear(OdeAdamsGear.Function function)

Constructs an ODE solver to solve the initial value problem dy/dt = f(t,y)

Parameters:

function - implementation of interface Function that defines the right-hand side function \(f(t,y)\)

Method Detail

setMaximumFunctionEvaluations

public void setMaximumFunctionEvaluations(int maxfcn)

Sets the maximum number of function evaluations of \(y'\) allowed.

Parameters:

maxfcn - an int specifying the maximum number of function evaluations of \(y'\) allowed. maxfcn must be greater than zero. Default: No limit is enforced.

setIntegrationMethod

public void setIntegrationMethod(int intMethod)

Indicates which integration method is to be used.

Parameters:

intMethod - an int specifying the integration method to be used. intMethod must be one of the values specified in the table which follows. Default: intMethod = METHOD_BDF

intMethod	Description
METHOD_ADAMS	Use the implicit Adams method.
METHOD_BDF	Use backward differentiation formula (BDF) methods.

setSolveMethod

public void setSolveMethod(int solveMethod)

Indicates which method to use for solving the formula equations.

Parameters:

solveMethod - an int specifying the method to be used for solving the formula equations. Note that if the problem is stiff and a chord or modified Newton method is most efficient, use SOLVE_CHORD_USER_JACOBIAN or SOLVE_CHORD_COMPUTED_JACOBIAN. solveMethod must be one of the values specified in the table which follows.
Default: solveMethod = SOLVE_CHORD_COMPUTED_JACOBIAN.

solveMethod	Description
SOLVE_FUNCTION_ITERATION	Use a function iteration or successive substitution method.
SOLVE_CHORD_USER_JACOBIAN	Use a chord or modified Newton method and a user-supplied Jacobian.
SOLVE_CHORD_COMPUTED_JACOBIAN	Use a chord or modified Newton method and a divided differences Jacobian.
SOLVE_CHORD_COMPUTED_DIAGONAL	Use a chord method and a diagonal matrix based on a directional directive.

setMaxOrder

public void setMaxOrder(int maxOrder)

Sets the highest order formula to use of implicit METHOD_ADAMS type or METHOD_BDF type.

Parameters:

maxOrder - an int specifying the highest order formula to use of implicit METHOD_ADAMS type or METHOD_BDF type. maxOrder must be greater than zero. Default: maxOrder = 12 for METHOD_ADAMS and maxOrder = 5 for METHOD_BDF.

setMaximumStepsize

public void setMaximumStepsize(double stepsize)

Sets the maximum internal step size.

Overrides:

setMaximumStepsize in class ODE

Parameters:

stepsize - a double specifying the maximum internal step size. stepsize must be greater than zero. Default: stepsize = 1.7976931348623157e+308

getNumberOfSteps

public int getNumberOfSteps()

Returns the number of internal steps taken.

Returns:

an int specifying the number of internal steps taken.

getMaximumFunctionEvaluations

public int getMaximumFunctionEvaluations()

Returns the maximum number of function evaluations of \(y'\) allowed.

Returns:

an int specifying the maximum number of function evaluations of \(y'\) allowed.

getNumberOfFcnEvals

public int getNumberOfFcnEvals()

Returns the number of function evaluations of \(y'\) made.

Returns:

an int specifying the number of function evaluations of \(y'\) made.

getNumberOfJacobianEvals

public int getNumberOfJacobianEvals()

Returns the number of Jacobian matrix evaluations used.

Returns:

an int specifying the number of Jacobian matrix evaluations used.

getIntegrationMethod

public int getIntegrationMethod()

Returns the integration method used.

Returns:

an int indicating the integration method used. One of the following is returned:

Value	Description
METHOD_ADAMS	Use the implicit Adams method.
METHOD_BDF	Use backward differentiation formula (BDF) methods.

getSolveMethod

public int getSolveMethod()

Returns the method for solving the formula equations.

Returns:

an int indicating the method used for solving the formula equations. One of the following is returned:

Value	Description
SOLVE_FUNCTION_ITERATION	Use a function iteration or successive substitution method.
SOLVE_CHORD_USER_JACOBIAN	Use a chord or modified Newton method and a user-supplied Jacobian.
SOLVE_CHORD_COMPUTED_JACOBIAN	Use a chord or modified Newton method and a Jacobian approximated by divided differences.
SOLVE_CHORD_COMPUTED_DIAGONAL	Use a chord method and a diagonal matrix based on a directional directive.

getMaxOrder

public int getMaxOrder()

Returns the highest order formula to use of implicit METHOD_ADAMS type or METHOD_BDF type.

Returns:

an int specifying the highest order formula to use of implicit METHOD_ADAMS type or METHOD_BDF type.

solve

public void solve(double t,
                  double tEnd,
                  double[] y)
           throws OdeAdamsGear.DidNotConvergeException,
                  OdeAdamsGear.MaxFcnEvalsExceededException,
                  OdeAdamsGear.ToleranceTooSmallException,
                  OdeAdamsGear.SingularMatrixException

Integrates the ODE system from t to tEnd. On all but the first call to solve, the value of t must equal the value of tEnd from the previous call.

Parameters:

t - a double specifying the independent variable

tEnd - a double specifying the value of t at which the solution is desired

y - on input, a double array containing the initial values. On output, a double array containing the approximate solution.

Throws:

OdeAdamsGear.DidNotConvergeException - is thrown if the number of internal steps exceeds maxSteps (default 500). This can be an indication that the ODE system is stiff. This exception can also be thrown if the error tolerance condition could not be met.

OdeAdamsGear.ToleranceTooSmallException - is thrown if the computation does not converge on some step.

OdeAdamsGear.MaxFcnEvalsExceededException - is thrown if the maximum number of function evaluations allowed has been exceeded.

SingularMatrixException - is thrown if the factorization function encounters a singular matrix during LU decomposition.

OdeAdamsGear.SingularMatrixException