org.apache.commons.math3.analysis.solvers

## Class UnivariateSolverUtils

- java.lang.Object
- - org.apache.commons.math3.analysis.solvers.UnivariateSolverUtils

-     public class UnivariateSolverUtils
      extends Object

  Utility routines for `UnivariateSolver` objects.
  Version:
    $Id: UnivariateSolverUtils.java 1579346 2014-03-19 18:43:39Z erans $

- - ### Method Summary

    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 was 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 NullArgumentException,
                                     NoBracketingException

      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:
        `NoBracketingException` - 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 NullArgumentException,
                                     NoBracketingException

      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:
        `NoBracketingException` - 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 NoBracketingException

      Force a root found by a non-bracketing solver to lie on a specified side, as if the solver was 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:
        `NoBracketingException` - if the function has the same sign at the endpoints.

    - #### bracket

          public static double[] bracket(UnivariateFunction function,
                         double initial,
                         double lowerBound,
                         double upperBound)
                                  throws NullArgumentException,
                                         NotStrictlyPositiveException,
                                         NoBracketingException

      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 `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.
      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:
        `NoBracketingException` - if a root cannot be bracketted.
        `NotStrictlyPositiveException` - if `maximumIterations <= 0`.
        `NullArgumentException` - if `function` is `null`.

    - #### bracket

          public static double[] bracket(UnivariateFunction function,
                         double initial,
                         double lowerBound,
                         double upperBound,
                         int maximumIterations)
                                  throws NullArgumentException,
                                         NotStrictlyPositiveException,
                                         NoBracketingException

      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:
        `NoBracketingException` - if the algorithm fails to find a and b satisfying the desired conditions.
        `NotStrictlyPositiveException` - 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 NoBracketingException

      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 limites -- NoBracketingException
      - `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 ({code 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 asymtotically 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:
        `NoBracketingException` - 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 NumberIsTooLargeException

      Check that the endpoints specify an interval.
      Parameters:
        `lower` - Lower endpoint.
        `upper` - Upper endpoint.
      Throws:
        `NumberIsTooLargeException` - if `lower >= upper`.

    - #### verifySequence

          public static void verifySequence(double lower,
                            double initial,
                            double upper)
                                     throws NumberIsTooLargeException

      Check that `lower < initial < upper`.
      Parameters:
        `lower` - Lower endpoint.
        `initial` - Initial value.
        `upper` - Upper endpoint.
      Throws:
        `NumberIsTooLargeException` - if `lower >= initial` or `initial >= upper`.

    - #### verifyBracketing

          public static void verifyBracketing(UnivariateFunction function,
                              double lower,
                              double upper)
                                       throws NullArgumentException,
                                              NoBracketingException

      Check that the endpoints specify an interval and the end points bracket a root.
      Parameters:
        `function` - Function.
        `lower` - Lower endpoint.
        `upper` - Upper endpoint.
      Throws:
        `NoBracketingException` - if the function has the same sign at the endpoints.
        `NullArgumentException` - if `function` is `null`.

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