UnivariateSolverUtils (Hipparchus 1.0 API)

java.lang.Object
- org.hipparchus.analysis.solvers.UnivariateSolverUtils

public class UnivariateSolverUtils
extends Object

Utility routines for UnivariateSolver objects.

Method Summary

All Methods Static Methods Concrete Methods
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`.

Methods inherited from class java.lang.Object
clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

- Method Detail
  - solve
```
public static double solve(UnivariateFunction function,
                           double x0,
                           double x1)
                    throws MathIllegalArgumentException,
                           NullArgumentException
```
    Convenience method to find a zero of a univariate real function. A default solver is used.
    
    Parameters:
    
    function - Function.
    
    x0 - Lower bound for the interval.
    
    x1 - Upper bound for the interval.
    
    Returns:
    
    a value where the function is zero.
    
    Throws:
    
    MathIllegalArgumentException - if the function has the same sign at the endpoints.
    
    NullArgumentException - if function is null.
  - solve
```
public static double solve(UnivariateFunction function,
                           double x0,
                           double x1,
                           double absoluteAccuracy)
                    throws MathIllegalArgumentException,
                           NullArgumentException
```
    Convenience method to find a zero of a univariate real function. A default solver is used.
    
    Parameters:
    
    function - Function.
    
    x0 - Lower bound for the interval.
    
    x1 - Upper bound for the interval.
    
    absoluteAccuracy - Accuracy to be used by the solver.
    
    Returns:
    
    a value where the function is zero.
    
    Throws:
    
    MathIllegalArgumentException - if the function has the same sign at the endpoints.
    
    NullArgumentException - if function is null.
  - forceSide
```
public static double forceSide(int maxEval,
                               UnivariateFunction f,
                               BracketedUnivariateSolver<UnivariateFunction> bracketing,
                               double baseRoot,
                               double min,
                               double max,
                               AllowedSolution allowedSolution)
                        throws MathIllegalArgumentException
```
    Force a root found by a non-bracketing solver to lie on a specified side, as if the solver were a bracketing one.
    
    Parameters:
    
    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 solve
    
    bracketing - bracketing solver to use for shifting the root
    
    baseRoot - original root found by a previous non-bracketing solver
    
    min - minimal bound of the search interval
    
    max - maximal bound of the search interval
    
    allowedSolution - the kind of solutions that the root-finding algorithm may accept as solutions.
    
    Returns:
    
    a root approximation, on the specified side of the exact root
    
    Throws:
    
    MathIllegalArgumentException - if the function has the same sign at the endpoints.
  - bracket
```
public static double[] bracket(UnivariateFunction function,
                               double initial,
                               double lowerBound,
                               double upperBound)
                        throws MathIllegalArgumentException,
                               NullArgumentException
```
    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.
    Note: this method can take Integer.MAX_VALUE iterations to throw a MathIllegalStateException. 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.
    
    Parameters:
    
    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).
    
    Returns:
    
    a two-element array holding a and b.
    
    Throws:
    
    MathIllegalArgumentException - if a root cannot be bracketted.
    
    MathIllegalArgumentException - if maximumIterations <= 0.
    
    NullArgumentException - if function is null.
  - bracket
```
public static double[] bracket(UnivariateFunction function,
                               double initial,
                               double lowerBound,
                               double upperBound,
                               int maximumIterations)
                        throws MathIllegalArgumentException,
                               NullArgumentException
```
    This method simply calls bracket(function, initial, lowerBound, upperBound, q, r, maximumIterations) with q and r set to 1.0.
    
    Parameters:
    
    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 perform
    
    Returns:
    
    a two element array holding a and b.
    
    Throws:
    
    MathIllegalArgumentException - if the algorithm fails to find a and b satisfying the desired conditions.
    
    MathIllegalArgumentException - if maximumIterations <= 0.
    
    NullArgumentException - if function is null.
  - bracket
```
public static double[] bracket(UnivariateFunction function,
                               double initial,
                               double lowerBound,
                               double upperBound,
                               double q,
                               double r,
                               int maximumIterations)
                        throws MathIllegalArgumentException
```
    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.
    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:
    - at least one positive and one negative value have been found -- success!
    - both endpoints have reached their respective limits -- MathIllegalArgumentException
    - maximumIterations iterations elapse -- MathIllegalArgumentException
    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.
    Parameters:
    
    function - function to check
    
    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).
    
    q - additive offset used to compute bounds sequence (must be strictly positive)
    
    r - multiplicative factor used to compute bounds sequence
    
    maximumIterations - Maximum number of iterations to perform
    
    Returns:
    
    a two element array holding the bracketing values.
    
    Throws:
    
    MathIllegalArgumentException - if function cannot be bracketed in the search interval
  - midpoint
```
public static double midpoint(double a,
                              double b)
```
    Compute the midpoint of two values.
    
    Parameters:
    
    a - first value.
    
    b - second value.
    
    Returns:
    
    the midpoint.
  - isBracketing
```
public static boolean isBracketing(UnivariateFunction function,
                                   double lower,
                                   double upper)
                            throws NullArgumentException
```
    Check whether the interval bounds bracket a root. That is, if the values at the endpoints are not equal to zero, then the function takes opposite signs at the endpoints.
    
    Parameters:
    
    function - Function.
    
    lower - Lower endpoint.
    
    upper - Upper endpoint.
    
    Returns:
    
    true if the function values have opposite signs at the given points.
    
    Throws:
    
    NullArgumentException - if function is null.
  - isSequence
```
public static boolean isSequence(double start,
                                 double mid,
                                 double end)
```
    Check whether the arguments form a (strictly) increasing sequence.
    
    Parameters:
    
    start - First number.
    
    mid - Second number.
    
    end - Third number.
    
    Returns:
    
    true if the arguments form an increasing sequence.
  - verifyInterval
```
public static void verifyInterval(double lower,
                                  double upper)
                           throws MathIllegalArgumentException
```
    Check that the endpoints specify an interval.
    
    Parameters:
    
    lower - Lower endpoint.
    
    upper - Upper endpoint.
    
    Throws:
    
    MathIllegalArgumentException - if lower >= upper.
  - verifySequence
```
public static void verifySequence(double lower,
                                  double initial,
                                  double upper)
                           throws MathIllegalArgumentException
```
    Check that lower < initial < upper.
    
    Parameters:
    
    lower - Lower endpoint.
    
    initial - Initial value.
    
    upper - Upper endpoint.
    
    Throws:
    
    MathIllegalArgumentException - if lower >= initial or initial >= upper.
  - verifyBracketing
```
public static void verifyBracketing(UnivariateFunction function,
                                    double lower,
                                    double upper)
                             throws MathIllegalArgumentException,
                                    NullArgumentException
```
    Check that the endpoints specify an interval and the end points bracket a root.
    
    Parameters:
    
    function - Function.
    
    lower - Lower endpoint.
    
    upper - Upper endpoint.
    
    Throws:
    
    MathIllegalArgumentException - if the function has the same sign at the endpoints.
    
    NullArgumentException - if function is null.

Class UnivariateSolverUtils

Method Summary

Methods inherited from class java.lang.Object

Method Detail

solve

solve

forceSide

bracket

bracket

bracket

midpoint

isBracketing

isSequence

verifyInterval

verifySequence

verifyBracketing