BSpline (Java Libraries for Stochastic Simulation)

java.lang.Object
- umontreal.iro.lecuyer.functionfit.BSpline

All Implemented Interfaces:

MathFunction, MathFunctionWithDerivative, MathFunctionWithFirstDerivative, MathFunctionWithIntegral
```
public class BSpline
extends java.lang.Object
implements MathFunction, MathFunctionWithIntegral, MathFunctionWithDerivative, MathFunctionWithFirstDerivative
```
Represents a B-spline with control points at (X_i, Y_i). Let P_i = (X_i, Y_i), for i = 0,…, n - 1, be a control point and let t_j, for j = 0,…, m - 1 be a knot. A B-spline of degree p = m - n - 1 is a parametric curve defined as

P(t) = ∑_i=0^n-1N_{i, p}(t)P_i, for t_p <= t <= t_m-p-1.
Here,

N_{i, p}(t) = ((t - t_i)/(t_i+p - t_i))N_{i, p-1}(t) + ((t_i+p+1 - t)/(t_i+p+1 - t_i+1))N_{i+1, p-1}(t)

N_{i, 0}(t) = { $\displaystyle \begin{array}{ll} 1&\mbox{ for }t_i\le t\le t_{i+1},\\ 0&\mbox{ elsewhere}. \end{array}$

This class provides methods to evaluate P(t) = (X(t), Y(t)) at any value of t, for a B-spline of any degree p >= 1. Note that the evaluate method of this class can be slow, since it uses a root finder to determine the value of t^* for which X(t^*) = x before it computes Y(t^*).

Constructor Summary

Constructors
Constructor and Description
`BSpline(double[] x, double[] y, double[] knots)` Constructs a new uniform B-spline with control points at (`x[i]`, `y[i]`), and knot vector given by the array `knots`.
`BSpline(double[] x, double[] y, int degree)` Constructs a new uniform B-spline of degree `degree` with control points at (`x[i]`, `y[i]`).

Method Summary

Methods
Modifier and Type	Method and Description
`static BSpline`	`createApproxBSpline(double[] x, double[] y, int degree, int h)` Returns a B-spline curve of degree `degree` smoothing (x_i, y_i), for i = 0,…, n points.
`static BSpline`	`createInterpBSpline(double[] x, double[] y, int degree)` Returns a B-spline curve of degree `degree` interpolating the (x_i, y_i) points.
`double`	`derivative(double u)` Computes (or estimates) the first derivative of the function at point `x`.
`double`	`derivative(double u, int n)` Computes (or estimates) the nth derivative of the function at point `x`.
`BSpline`	`derivativeBSpline()` Returns the derivative B-spline object of the current variable.
`BSpline`	`derivativeBSpline(int i)` Returns the ith derivative B-spline object of the current variable; i must be less than the degree of the original B-spline.
`double`	`evaluate(double u)` Returns the value of the function evaluated at x.
`double`	`evalX(double u)`
`double`	`evalY(double u)`
`double[]`	`getKnots()` Returns an array containing the knot vector (t₀, t_m-1).
`double`	`getMaxKnot()` Returns the knot maximal value.
`double`	`getMinKnot()` Returns the knot minimal value.
`double[]`	`getX()` Returns the X_i coordinates for this spline.
`double[]`	`getY()` Returns the Y_i coordinates for this spline.
`double`	`integral(double a, double b)` Computes (or estimates) the integral of the function over the interval [a, b].

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

- Constructor Detail
  - BSpline
```
public BSpline(double[] x,
       double[] y,
       int degree)
```
    Constructs a new uniform B-spline of degree degree with control points at (x[i], y[i]). The knots of the resulting B-spline are set uniformly from x[0] to x[n-1].
    
    Parameters:
    x - the values of X.
    y - the values of Y.
    degree - the degree of the B-spline.
  - BSpline
```
public BSpline(double[] x,
       double[] y,
       double[] knots)
```
    Constructs a new uniform B-spline with control points at (x[i], y[i]), and knot vector given by the array knots.
    
    Parameters:
    x - the values of X.
    y - the values of Y.
    knots - the knots of the B-spline.
- Method Detail
  - getX
```
public double[] getX()
```
    Returns the X_i coordinates for this spline.
    
    Returns:
    the X_i coordinates.
  - getY
```
public double[] getY()
```
    Returns the Y_i coordinates for this spline.
    
    Returns:
    the Y_i coordinates.
  - getMaxKnot
```
public double getMaxKnot()
```
    Returns the knot maximal value.
    
    Returns:
    the Y_i coordinates.
  - getMinKnot
```
public double getMinKnot()
```
    Returns the knot minimal value.
    
    Returns:
    the Y_i coordinates.
  - getKnots
```
public double[] getKnots()
```
    Returns an array containing the knot vector (t₀, t_m-1).
    
    Returns:
    the knot vector.
  - createInterpBSpline
```
public static BSpline createInterpBSpline(double[] x,
                          double[] y,
                          int degree)
```
    Returns a B-spline curve of degree degree interpolating the (x_i, y_i) points. This method uses the uniformly spaced method for interpolating points with a B-spline curve, and a uniformed clamped knot vector, as described in http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/.
    
    Parameters:
    x - the values of X.
    y - the values of Y.
    degree - the degree of the B-spline.
    
    Returns:
    the B-spline curve.
  - createApproxBSpline
```
public static BSpline createApproxBSpline(double[] x,
                          double[] y,
                          int degree,
                          int h)
```
    Returns a B-spline curve of degree degree smoothing (x_i, y_i), for i = 0,…, n points. The precision depends on the parameter h: 1 <= degree <= h < n, which represents the number of control points used by the new B-spline curve, minimizing the quadratic error
    
    L = ∑_i=0ⁿ( $\displaystyle {\frac{{Y_i - S_i(X_i)}}{{W_i}}}$ )².
    This method uses the uniformly spaced method for interpolating points with a B-spline curve and a uniformed clamped knot vector, as described in http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/.
    
    Parameters:
    x - the values of X.
    y - the values of Y.
    degree - the degree of the B-spline.
    h - the desired number of control points.
    
    Returns:
    the B-spline curve.
  - derivativeBSpline
```
public BSpline derivativeBSpline()
```
    Returns the derivative B-spline object of the current variable. Using this function and the returned object, instead of the derivative method, is strongly recommended if one wants to compute many derivative values.
    
    Returns:
    the derivative B-spline of the current variable.
  - derivativeBSpline
```
public BSpline derivativeBSpline(int i)
```
    Returns the ith derivative B-spline object of the current variable; i must be less than the degree of the original B-spline. Using this function and the returned object, instead of the derivative method, is strongly recommended if one wants to compute many derivative values.
    
    Parameters:
    i - the degree of the derivative.
    
    Returns:
    the ith derivative.
  - evaluate
```
public double evaluate(double u)
```
    Description copied from interface: MathFunction
    
    Returns the value of the function evaluated at x.
    
    Specified by:
    
    evaluate in interface MathFunction
    
    Parameters:
    u - value at which the distribution function is evaluated
    
    Returns:
    function evaluated at x
  - integral
```
public double integral(double a,
              double b)
```
    Description copied from interface: MathFunctionWithIntegral
    
    Computes (or estimates) the integral of the function over the interval [a, b].
    
    Specified by:
    
    integral in interface MathFunctionWithIntegral
    
    Parameters:
    a - the starting point of the interval.
    b - the ending point of the interval.
    
    Returns:
    the value of the integral.
  - derivative
```
public double derivative(double u)
```
    Description copied from interface: MathFunctionWithFirstDerivative
    
    Computes (or estimates) the first derivative of the function at point x.
    
    Specified by:
    
    derivative in interface MathFunctionWithFirstDerivative
    
    Parameters:
    u - the point to evaluate the derivative to.
    
    Returns:
    the value of the derivative.
  - derivative
```
public double derivative(double u,
                int n)
```
    Description copied from interface: MathFunctionWithDerivative
    
    Computes (or estimates) the nth derivative of the function at point x. For n = 0, this returns the result of evaluate.
    
    Specified by:
    
    derivative in interface MathFunctionWithDerivative
    
    Parameters:
    u - the point to evaluate the derivate to.
    n - the order of the derivative.
    
    Returns:
    the resulting derivative.
  - evalX
```
public double evalX(double u)
```
  - evalY
```
public double evalY(double u)
```

N_{i, p}(t)	=	((t - t_i)/(t_i+p - t_i))N_{i, p-1}(t) + ((t_i+p+1 - t)/(t_i+p+1 - t_i+1))N_{i+1, p-1}(t)
N_{i, 0}(t)	=	{ $\displaystyle \begin{array}{ll} 1&\mbox{ for }t_i\le t\le t_{i+1},\\ 0&\mbox{ elsewhere}. \end{array}$

Class BSpline

Constructor Summary

Method Summary

Methods inherited from class java.lang.Object

Constructor Detail

BSpline

BSpline

Method Detail

getX

getY

getMaxKnot

getMinKnot

getKnots

createInterpBSpline

createApproxBSpline

derivativeBSpline

derivativeBSpline

evaluate

integral

derivative

derivative

evalX

evalY