Packages

  • package root
    Definition Classes
    root
  • package scalation

    The scalation package specifies system-wide constants for directory paths.

    The scalation package specifies system-wide constants for directory paths. Sub-packages may wish to define 'BASE-DIR = DATA_DIR + ⁄ + <package>' in their own 'package.scala' files. For maintainability, directory paths should only be specified in 'package.scala' files.

    Definition Classes
    root
  • package calculus

    The calculus package contains classes with methods for computing derivatives, gradient vectors, Jacobian matrices, integrals and basic operators in Functional Analysis.

    The calculus package contains classes with methods for computing derivatives, gradient vectors, Jacobian matrices, integrals and basic operators in Functional Analysis.

    Definition Classes
    scalation
  • Differential
  • DifferentialTest
  • DifferentialTest2
  • GaussianFunc
  • Hilbert
  • HilbertTest
  • Integral
  • IntegralTest
  • IntegralTest2
o

scalation.calculus

Differential

object Differential

The Differential object contains functions for computing derivatives, partial derivatives, Laplacians, gradient vectors, Hessian matrices and Jacobian matrices.

Linear Supertypes
AnyRef, Any
Ordering
  1. Alphabetic
  2. By Inheritance
Inherited
  1. Differential
  2. AnyRef
  3. Any
  1. Hide All
  2. Show All
Visibility
  1. Public
  2. All

Type Members

  1. type FunctionV2S = (VectorD) ⇒ Double
  2. type FunctionV_2S = (VectoD) ⇒ Double
  3. type FunctionsV = Array[FunctionV2S]

Value Members

  1. final def !=(arg0: Any): Boolean
    Definition Classes
    AnyRef → Any
  2. final def ##(): Int
    Definition Classes
    AnyRef → Any
  3. final def ==(arg0: Any): Boolean
    Definition Classes
    AnyRef → Any
  4. final def asInstanceOf[T0]: T0
    Definition Classes
    Any
  5. def clone(): AnyRef
    Attributes
    protected[java.lang]
    Definition Classes
    AnyRef
    Annotations
    @throws( ... )
  6. def derivative(f: FunctionS2S, x: Double): Double

    Estimate the derivative of the scalar-to-scalar function 'f' at 'x' using a 2-sided method (central difference).

    Estimate the derivative of the scalar-to-scalar function 'f' at 'x' using a 2-sided method (central difference). Approximate the tangent line at '(x, f(x))' with the secant line through points '(x-h, f(x-h))' and '(x+h, f(x+h))'. Tends to be MORE ACCURATE than the 1-sided method.

    f

    the function whose derivative is sought

    x

    the point (scalar) at which to estimate the derivative

    See also

    www.math.montana.edu/frankw/ccp/modeling/continuous/heatflow2/firstder.htm

  7. def derivative1(f: FunctionS2S, x: Double): Double

    Estimate the derivative of the scalar-to-scalar function 'f' at 'x' using a 1-sided method (forward difference).

    Estimate the derivative of the scalar-to-scalar function 'f' at 'x' using a 1-sided method (forward difference). Approximate the tangent line at '(x, f(x))' with the secant line through points '(x, f(x))' and '(x+h, f(x+h))'.

    f

    the function whose derivative is sought

    x

    the point (scalar) at which to estimate the derivative

  8. def derivative2(f: FunctionS2S, x: Double): Double

    Estimate the second derivative of the scalar-to-scalar function 'f' at 'x' using the central difference formula for second derivatives.

    Estimate the second derivative of the scalar-to-scalar function 'f' at 'x' using the central difference formula for second derivatives.

    f

    the function whose second derivative is sought

    x

    the point (scalar) at which to estimate the derivative

  9. final def eq(arg0: AnyRef): Boolean
    Definition Classes
    AnyRef
  10. def equals(arg0: Any): Boolean
    Definition Classes
    AnyRef → Any
  11. def finalize(): Unit
    Attributes
    protected[java.lang]
    Definition Classes
    AnyRef
    Annotations
    @throws( classOf[java.lang.Throwable] )
  12. final def getClass(): Class[_]
    Definition Classes
    AnyRef → Any
  13. def gradient(f: FunctionV2S, x: VectorD): VectorD

    Estimate the gradient of the vector-to-scalar function 'f' at point 'x' returning a value for the partial derivative for each dimension of 'x'.

    Estimate the gradient of the vector-to-scalar function 'f' at point 'x' returning a value for the partial derivative for each dimension of 'x'.

    f

    the function whose gradient is sought

    x

    the point (vector) at which to estimate the gradient

  14. def gradientD(d: FunctionsV, x: VectorD): VectorD

    Compute the gradient of the vector-to-scalar function 'f' using partial derivative functions evaluated at point 'x.' Return a value for the partial derivative for each dimension of the vector 'x.'

    Compute the gradient of the vector-to-scalar function 'f' using partial derivative functions evaluated at point 'x.' Return a value for the partial derivative for each dimension of the vector 'x.'

    d

    the array of partial derivative functions

    x

    the point (vector) at which to compute the gradient

  15. def hashCode(): Int
    Definition Classes
    AnyRef → Any
  16. def hessian(f: FunctionV2S, x: VectorD): MatrixD

    Estimate the Hessian of the vector-to-scalar function 'f' at point 'x' returning a matrix of second partial derivative.

    Estimate the Hessian of the vector-to-scalar function 'f' at point 'x' returning a matrix of second partial derivative.

    f

    the function whose Hessian is sought

    x

    the point (vector) at which to estimate the Hessian

  17. final def isInstanceOf[T0]: Boolean
    Definition Classes
    Any
  18. def jacobian(f: FunctionsV, x: VectorD): MatrixD

    Compute the Jacobian matrix for a vector-valued function represented as an array of scalar-valued functions.

    Compute the Jacobian matrix for a vector-valued function represented as an array of scalar-valued functions. The 'i'th row in the matrix is the gradient of the 'i'th function.

    f

    the array of functions whose Jacobian is sought

    x

    the point (vector) at which to estimate the Jacobian

  19. def laplacian(f: FunctionV2S, x: VectorD): Double

    Estimate the Laplacian of the vector-to-scalar function 'f' at point 'x' returning the sum of the pure second partial derivatives.

    Estimate the Laplacian of the vector-to-scalar function 'f' at point 'x' returning the sum of the pure second partial derivatives.

    f

    the function whose Hessian is sought

    x

    the point (vector) at which to estimate the Hessian

  20. final def ne(arg0: AnyRef): Boolean
    Definition Classes
    AnyRef
  21. final def notify(): Unit
    Definition Classes
    AnyRef
  22. final def notifyAll(): Unit
    Definition Classes
    AnyRef
  23. def partial(i: Int)(f: FunctionV2S, x: VectorD): Double

    Estimate the 'i'th partial derivative of the vector-to-scalar function 'f' at point 'x' returning the value for the partial derivative for dimension 'i'.

    Estimate the 'i'th partial derivative of the vector-to-scalar function 'f' at point 'x' returning the value for the partial derivative for dimension 'i'.

    i

    the dimension to compute the partial derivative on

    f

    the function whose partial derivative is sought

    x

    the point (vector) at which to estimate the partial derivative

  24. def partial2(i: Int, j: Int)(f: FunctionV2S, x: VectorD): Double

    Estimate the '(i,j)'th second partial derivative of the vector-to-scalar function 'f' at point 'x' returning the value for the second partial derivative for dimensions '(i, j)'.

    Estimate the '(i,j)'th second partial derivative of the vector-to-scalar function 'f' at point 'x' returning the value for the second partial derivative for dimensions '(i, j)'. If 'i = j', the second partial derivative is called "pure", otherwise it is a "cross" second partial derivative.

    i

    the first dimension to compute the second partial derivative on

    j

    the second dimension to compute the second partial derivative on

    f

    the function whose second partial derivative is sought

    x

    the point (vector) at which to estimate the second partial derivative

    See also

    www.uio.no/studier/emner/matnat/math/MAT-INF1100/h07/undervisningsmateriale/kap7.pdf

  25. def resetH(step: Double): Unit

    Reset the step size from its default step size to one more suitable for your function.

    Reset the step size from its default step size to one more suitable for your function. A heuristic for the central difference method is to let 'h = max (|x|,1) * (machine-epsilon)^(1/3)' For double precision, the machine-epsilon is about 1E-16.

    step

    the new step size to reset h to

    See also

    www.karenkopecky.net/Teaching/eco613614/Notes_NumericalDifferentiation.pdf

  26. def slope(f: FunctionV2S, x: VectorD, n: Int = 0): VectorD

    Compute the slope of the vector-to-scalar function 'f' defined on mixed real/integer vectors.

    Compute the slope of the vector-to-scalar function 'f' defined on mixed real/integer vectors.

    f

    the function whose slope is sought

    x

    the point (vector) at which to estimate the slope

    n

    the number of dimensions that are real-valued (rest are integers)

  27. final def synchronized[T0](arg0: ⇒ T0): T0
    Definition Classes
    AnyRef
  28. def toString(): String
    Definition Classes
    AnyRef → Any
  29. final def wait(): Unit
    Definition Classes
    AnyRef
    Annotations
    @throws( ... )
  30. final def wait(arg0: Long, arg1: Int): Unit
    Definition Classes
    AnyRef
    Annotations
    @throws( ... )
  31. final def wait(arg0: Long): Unit
    Definition Classes
    AnyRef
    Annotations
    @throws( ... )
  32. def Η(f: FunctionV2S, x: VectorD): MatrixD
  33. def Ј(f: FunctionsV, x: VectorD): MatrixD
  34. def (f: FunctionS2S, x: Double): Double
  35. def ⅮⅮ(f: FunctionS2S, x: Double): Double
  36. def (i: Int)(f: FunctionV2S, x: VectorD): Double
  37. def ∂∂(i: Int, j: Int)(f: FunctionV2S, x: VectorD): Double
  38. def (f: FunctionV2S, x: VectorD): Double
  39. def (f: FunctionV2S, x: VectorD): VectorD
  40. def ∇*(d: FunctionsV, x: VectorD): VectorD

Inherited from AnyRef

Inherited from Any

Ungrouped