public class UnivariateSolverUtils extends Object
UnivariateSolver objects.| Modifier and Type | Method and Description |
|---|---|
static double[] |
bracket(UnivariateFunction function,
double initial,
double lowerBound,
double upperBound)
This method simply calls
bracket(function, initial, lowerBound, upperBound, q, r, maximumIterations)
with q and r set to 1.0 and maximumIterations set to Integer.MAX_VALUE. |
static double[] |
bracket(UnivariateFunction function,
double initial,
double lowerBound,
double upperBound,
double q,
double r,
int maximumIterations)
This method attempts to find two values a and b satisfying
lowerBound <= a < initial < b <= upperBound
f(a) * f(b) <= 0
If f is continuous on [a,b], this means that a
and b bracket a root of f. |
static double[] |
bracket(UnivariateFunction function,
double initial,
double lowerBound,
double upperBound,
int maximumIterations)
This method simply calls
bracket(function, initial, lowerBound, upperBound, q, r, maximumIterations)
with q and r set to 1.0. |
static double |
forceSide(int maxEval,
UnivariateFunction f,
BracketedUnivariateSolver<UnivariateFunction> bracketing,
double baseRoot,
double min,
double max,
AllowedSolution allowedSolution)
Force a root found by a non-bracketing solver to lie on a specified side,
as if the solver were a bracketing one.
|
static boolean |
isBracketing(UnivariateFunction function,
double lower,
double upper)
Check whether the interval bounds bracket a root.
|
static boolean |
isSequence(double start,
double mid,
double end)
Check whether the arguments form a (strictly) increasing sequence.
|
static double |
midpoint(double a,
double b)
Compute the midpoint of two values.
|
static double |
solve(UnivariateFunction function,
double x0,
double x1)
Convenience method to find a zero of a univariate real function.
|
static double |
solve(UnivariateFunction function,
double x0,
double x1,
double absoluteAccuracy)
Convenience method to find a zero of a univariate real function.
|
static void |
verifyBracketing(UnivariateFunction function,
double lower,
double upper)
Check that the endpoints specify an interval and the end points
bracket a root.
|
static void |
verifyInterval(double lower,
double upper)
Check that the endpoints specify an interval.
|
static void |
verifySequence(double lower,
double initial,
double upper)
Check that
lower < initial < upper. |
public static double solve(UnivariateFunction function, double x0, double x1) throws NullArgumentException, NoBracketingException
function - Function.x0 - Lower bound for the interval.x1 - Upper bound for the interval.NoBracketingException - if the function has the same sign at the
endpoints.NullArgumentException - if function is null.public static double solve(UnivariateFunction function, double x0, double x1, double absoluteAccuracy) throws NullArgumentException, NoBracketingException
function - Function.x0 - Lower bound for the interval.x1 - Upper bound for the interval.absoluteAccuracy - Accuracy to be used by the solver.NoBracketingException - if the function has the same sign at the
endpoints.NullArgumentException - if function is null.public static double forceSide(int maxEval,
UnivariateFunction f,
BracketedUnivariateSolver<UnivariateFunction> bracketing,
double baseRoot,
double min,
double max,
AllowedSolution allowedSolution)
throws NoBracketingException
maxEval - maximal number of new evaluations of the function
(evaluations already done for finding the root should have already been subtracted
from this number)f - function to solvebracketing - bracketing solver to use for shifting the rootbaseRoot - original root found by a previous non-bracketing solvermin - minimal bound of the search intervalmax - maximal bound of the search intervalallowedSolution - the kind of solutions that the root-finding algorithm may
accept as solutions.NoBracketingException - if the function has the same sign at the
endpoints.public static double[] bracket(UnivariateFunction function, double initial, double lowerBound, double upperBound) throws NullArgumentException, NotStrictlyPositiveException, NoBracketingException
bracket(function, initial, lowerBound, upperBound, q, r, maximumIterations)
with q and r set to 1.0 and maximumIterations set to Integer.MAX_VALUE.
Note: this method can take Integer.MAX_VALUE
iterations to throw a ConvergenceException. Unless you are
confident that there is a root between lowerBound and
upperBound near initial, it is better to use
bracket(function, initial, lowerBound, upperBound, q, r, maximumIterations),
explicitly specifying the maximum number of iterations.
function - Function.initial - Initial midpoint of interval being expanded to
bracket a root.lowerBound - Lower bound (a is never lower than this value)upperBound - Upper bound (b never is greater than this
value).NoBracketingException - if a root cannot be bracketted.NotStrictlyPositiveException - if maximumIterations <= 0.NullArgumentException - if function is null.public static double[] bracket(UnivariateFunction function, double initial, double lowerBound, double upperBound, int maximumIterations) throws NullArgumentException, NotStrictlyPositiveException, NoBracketingException
bracket(function, initial, lowerBound, upperBound, q, r, maximumIterations)
with q and r set to 1.0.function - Function.initial - Initial midpoint of interval being expanded to
bracket a root.lowerBound - Lower bound (a is never lower than this value).upperBound - Upper bound (b never is greater than this
value).maximumIterations - Maximum number of iterations to performNoBracketingException - if the algorithm fails to find a and b
satisfying the desired conditions.NotStrictlyPositiveException - if maximumIterations <= 0.NullArgumentException - if function is null.public static double[] bracket(UnivariateFunction function, double initial, double lowerBound, double upperBound, double q, double r, int maximumIterations) throws NoBracketingException
lowerBound <= a < initial < b <= upperBound f(a) * f(b) <= 0 f is continuous on [a,b], this means that a
and b bracket a root of f.
The algorithm checks the sign of \( f(l_k) \) and \( f(u_k) \) for increasing values of k, where \( l_k = max(lower, initial - \delta_k) \), \( u_k = min(upper, initial + \delta_k) \), using recurrence \( \delta_{k+1} = r \delta_k + q, \delta_0 = 0\) and starting search with \( k=1 \). The algorithm stops when one of the following happens:
maximumIterations iterations elapse -- NoBracketingException
If different signs are found at first iteration (k=1), then the returned
interval will be \( [a, b] = [l_1, u_1] \). If different signs are found at a later
iteration k>1, then the returned interval will be either
\( [a, b] = [l_{k+1}, l_{k}] \) or \( [a, b] = [u_{k}, u_{k+1}] \). A root solver called
with these parameters will therefore start with the smallest bracketing interval known
at this step.
Interval expansion rate is tuned by changing the recurrence parameters r and
q. When the multiplicative factor r is set to 1, the sequence is a
simple arithmetic sequence with linear increase. When the multiplicative factor r
is larger than 1, the sequence has an asymptotically exponential rate. Note than the
additive parameter q should never be set to zero, otherwise the interval would
degenerate to the single initial point for all values of k.
As a rule of thumb, when the location of the root is expected to be approximately known
within some error margin, r should be set to 1 and q should be set to the
order of magnitude of the error margin. When the location of the root is really a wild guess,
then r should be set to a value larger than 1 (typically 2 to double the interval
length at each iteration) and q should be set according to half the initial
search interval length.
As an example, if we consider the trivial function f(x) = 1 - x and use
initial = 4, r = 1, q = 2, the algorithm will compute
f(4-2) = f(2) = -1 and f(4+2) = f(6) = -5 for k = 1, then
f(4-4) = f(0) = +1 and f(4+4) = f(8) = -7 for k = 2. Then it will
return the interval [0, 2] as the smallest one known to be bracketing the root.
As shown by this example, the initial value (here 4) may lie outside of the returned
bracketing interval.
function - function to checkinitial - Initial midpoint of interval being expanded to
bracket a root.lowerBound - Lower bound (a is never lower than this value).upperBound - Upper bound (b never is greater than this
value).q - additive offset used to compute bounds sequence (must be strictly positive)r - multiplicative factor used to compute bounds sequencemaximumIterations - Maximum number of iterations to performNoBracketingException - if function cannot be bracketed in the search intervalpublic static double midpoint(double a,
double b)
a - first value.b - second value.public static boolean isBracketing(UnivariateFunction function, double lower, double upper) throws NullArgumentException
function - Function.lower - Lower endpoint.upper - Upper endpoint.true if the function values have opposite signs at the
given points.NullArgumentException - if function is null.public static boolean isSequence(double start,
double mid,
double end)
start - First number.mid - Second number.end - Third number.true if the arguments form an increasing sequence.public static void verifyInterval(double lower,
double upper)
throws NumberIsTooLargeException
lower - Lower endpoint.upper - Upper endpoint.NumberIsTooLargeException - if lower >= upper.public static void verifySequence(double lower,
double initial,
double upper)
throws NumberIsTooLargeException
lower < initial < upper.lower - Lower endpoint.initial - Initial value.upper - Upper endpoint.NumberIsTooLargeException - if lower >= initial or
initial >= upper.public static void verifyBracketing(UnivariateFunction function, double lower, double upper) throws NullArgumentException, NoBracketingException
function - Function.lower - Lower endpoint.upper - Upper endpoint.NoBracketingException - if the function has the same sign at the
endpoints.NullArgumentException - if function is null.Copyright © 2003–2016 The Apache Software Foundation. All rights reserved.