calculus

package calculus

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

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class B_Spline extends BasisFunction with Error
The B_Spline class provides B-Spline basis functions for various orders 'm', where the order is one more than the degree.
The B_Spline class provides B-Spline basis functions for various orders 'm', where the order is one more than the degree. A spline function is a piecewise polynomial function where the pieces are stitched together at knots with the goal of maintaining continuity and differentiability. B-Spline basis functions form a popular form of basis functions used in Functional Data Analysis.

See also
http://web.mit.edu/hyperbook/Patrikalakis-Maekawa-Cho/node17.html -----------------------------------------------------------------------------
trait BasisFunction extends AnyRef
The BasisFunction trait provides a common framework for various Basis Functions.
The BasisFunction trait provides a common framework for various Basis Functions. They are a set of functions forming a basis whose orthogonal components form a function space. Two functions 'f' and 'g' are orthogonal if their inner product is 0, meaning <f, g> = 0.
class DB_Spline extends B_Spline with DBasisFunction
The DB_Spline class provides B-Spline basis functions with derivatives for various orders 'm', where the order is one more than the degree.
The DB_Spline class provides B-Spline basis functions with derivatives for various orders 'm', where the order is one more than the degree. A spline function is a piecewise polynomial function where the pieces are stitched together at knots with the goal of maintaining continuity and differentability. B-Spline basis functions form a popular form of basis functions used in Functional Data Analysis.

See also
http://web.mit.edu/hyperbook/Patrikalakis-Maekawa-Cho/node17.html -----------------------------------------------------------------------------
trait DBasisFunction extends BasisFunction
The DBasisFunction trait provides a common framework for the derivatives of various Basis Functions.
class DFourier extends Fourier with DBasisFunction
The DFourier class provides Fourier basis functions with derivatives.
The DFourier class provides Fourier basis functions with derivatives. Such basis functions are useful are useful for fitting periodic data in Functional Data Analysis.

See also
en.wikipedia.org/wiki/Fourier_series
class DRadial extends Radial with DBasisFunction
The DRadial class provides Radial basis functions with derivatives.
The DRadial class provides Radial basis functions with derivatives. Such basis functions are useful are useful in Neural Networks and Support Vector Classification.
class Fourier extends BasisFunction with Error
The Fourier class provides Fourier basis functions.
The Fourier class provides Fourier basis functions. Such basis functions are useful are useful for fitting periodic data in Functional Data Analysis.

See also
en.wikipedia.org/wiki/Fourier_series
class GaussianFunc extends AnyRef
The GaussianFunc class implements the Gaussian function, a generalization of the Gaussian/Normal distribution density function.
The GaussianFunc class implements the Gaussian function, a generalization of the Gaussian/Normal distribution density function.

See also
en.wikipedia.org/wiki/Gaussian_function
class Hilbert extends AnyRef
The Hilbert class provides operators to add, subtract, mutiply, divide and raise functions.
The Hilbert class provides operators to add, subtract, mutiply, divide and raise functions. Given two functions, 'f' and 'g', a new function is created. It also provides methods for computing dot/inner products, norms and distances for functions defined in Hilbert Space. On interval [a, b]
Lp-norm (f) = [ ∫f(t)^{p dt ]}1/p

See also
implicit conversion 'functionS2S2Hilbert' in package.scala
type Interval = (Double, Double)
Type definition for an interval [a, b]
case class Poly(c: VectorD, x: String = "x") extends Product with Serializable
The Poly class provides operations on univariate polynomials.
The Poly class provides operations on univariate polynomials.
Poly (2, 3) => 3 x + 2
Note, reverse order of coefficients, i.e., coefficients for smallest terms first.
c
the coefficients of the polynomial
x
the variable/indeterminate of the polynomial

See also
MPoly' for multivariate polynomials.
class Radial extends BasisFunction with Error
The Radial class provides Radial basis functions.
The Radial class provides Radial basis functions. Such basis functions are useful are useful in Neural Networks and Support Vector Classification. -----------------------------------------------------------------------------

Value Members

def _0f(x: Double): Double
Zero function.
def _1f(x: Double): Double
One function.
implicit def functionS2S2Hilbert(f: FunctionS2S): Hilbert
Implicit conversion from 'FunctionS2S' to 'Hilbert', which supports functional operators.
Implicit conversion from 'FunctionS2S' to 'Hilbert', which supports functional operators.
f
the function to turn into a Hilbert function
object B_Spline
Companion object for B_Spline class provides functions for clamping the the ends of a spline and running timing benchmarks.
object B_SplineTest extends App
The B_SplineTest object is used to test the B_Spline class.
The B_SplineTest object is used to test the B_Spline class. It tests the B-Spline functions using the general recurrence. > runMain scalation.calculus.B_SplineTest
object B_SplineTest2 extends App
The B_SplineTest2 object is used to test the B_Spline class.
The B_SplineTest2 object is used to test the B_Spline class. It tests the B-Spline functions using the general recurrence and plots several basis functions using PlotM. > runMain scalation.calculus.B_SplineTest2
object B_SplineTest3 extends App
The B_SplineTest3 object is used to test the B_Spline class.
The B_SplineTest3 object is used to test the B_Spline class. It tests the B-Spline functions using the general recurrence. > runMain scalation.calculus.B_SplineTest3
object B_SplineTest4 extends App
The B_SplineTest4 object is used to test the B_Spline class.
The B_SplineTest4 object is used to test the B_Spline class. It tests the B-Spline functions using the general recurrence. > runMain scalation.calculus.B_SplineTest4
object BasisFunction
The BasisFunction object provides utility functions related to basis functions.
object DB_SplineTest extends App
The DB_SplineTest object is used to test the DB_Spline class.
The DB_SplineTest object is used to test the DB_Spline class. Here, we compute the penalty matrix for a ridge regression. > runMain scalation.caculus.DB_SplineTest
object DB_SplineTest2 extends App
The DB_SplineTest2 object is used to test the DB_Spline class.
The DB_SplineTest2 object is used to test the DB_Spline class. Here, we smooth functional data manually with a roughness penalty.

See also
Ramsay et al. page 86 > runMain scalation.caculus.DB_SplineTest2
object DBasisFunction
The DBasisFunction object provides utility functions related to derivatives of basis functions.
object DFourierTest extends App
The DFourierTest object is used to test the DFourier class.
The DFourierTest object is used to test the DFourier class. > runMain scalation.calculus.DFourierTest
object DRadialTest extends App
The DRadialTest object is used to test the DRadial class.
The DRadialTest object is used to test the DRadial class. > runMain scalation.calculus.DRadialTest
object Differential
The Differential object contains functions for computing derivatives, partial derivatives, Laplacians, gradient vectors, Hessian matrices and Jacobian matrices.
object DifferentialTest extends App
The DifferentialTest object is used to test the Differential object.
The DifferentialTest object is used to test the Differential object. > runMain scalation.calculus.DifferentialTest
object DifferentialTest2 extends App
The DifferentialTest2 object is used to test the Differential object showing trade-offs of using 1-sided and 2-sided derivative approximations as well as different values for h.
The DifferentialTest2 object is used to test the Differential object showing trade-offs of using 1-sided and 2-sided derivative approximations as well as different values for h.

See also
www.rose-hulman.edu/~bryan/lottamath/diffgrad.pdf > runMain scalation.calculus.DifferentialTest2
object FFT
The FFT object provides a method to transform a signal in the time domain (e.g., a times series) in its representation in the frequency domain.
The FFT object provides a method to transform a signal in the time domain (e.g., a times series) in its representation in the frequency domain. The inverse transform is also provided.
object FFTTest extends App
The FFTTest is used to test the FFT object by starting with a signal in th4e time domain, converting it to its frequency domain representation and the converting it back to the time domain.
The FFTTest is used to test the FFT object by starting with a signal in th4e time domain, converting it to its frequency domain representation and the converting it back to the time domain. > runMain scalation.calculus.FFTTest
object FourierTest extends App
The FourierTest object is used to test the Fourier class.
The FourierTest object is used to test the Fourier class. > runMain scalation.calculus.FourierTest
object HilbertTest extends App
The HilbertTest object is used to test the Hilbert class.
The HilbertTest object is used to test the Hilbert class. > runMain scalation.calculus.HilbertTest
object Integral
The Integral object provides implementations for five basic integration methods:
The Integral object provides implementations for five basic integration methods:
∫f(x)dx on interval [a, b]
trap - trapezoidal method - linear simpson - Simpson method - quadratic simpson38 - 3/8 Simpson method - cubic boole - Boole Method - quartic romberg - Romberg method - recursive, uses trap
The first four are Composite Newton-Coates type integrators.

See also
en.wikipedia.org/wiki/Newton%E2%80%93Cotes_formulas
object IntegralTest extends App
The IntegralTest2 object tests the numerical integrators on simple problems.
The IntegralTest2 object tests the numerical integrators on simple problems. Easy problems. > runMain scalation.calculus.IntegralTest
object IntegralTest2 extends App
The IntegralTest2 object tests the numerical integrators using the Gauss function that has no analytic solution.
The IntegralTest2 object tests the numerical integrators using the Gauss function that has no analytic solution. Hard problem. > runMain scalation.calculus.IntegralTest2
object Poly extends Serializable
The Poly companion object provides factory methods for the 'Poly' class.
object PolyTest extends App
The PolyTest object is used to test the Poly class.
The PolyTest object is used to test the Poly class. > runMain scalation.calculus.PolyTest
object RadialTest extends App
The RadialTest object is used to test the Radial class.
The RadialTest object is used to test the Radial class. > runMain scalation.calculus.RadialTest
object RadialType extends Enumeration
Enumerations defining various Basis Functions to be used

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