1             Algorithmen in SimplexNumerica  

Hier nur eine Auswahl an Algorithmen:

 

1.1     Funktions-Plot

 

·         Regular Function f(x)

·         Parametric Function x(t), y(t)

·         Polar Function r(t)

·         Relations (Equation/Inequality)

·         Add Tangent

·         Add Normal

·         Fill Area

·         Custom Function Constants

 

1.2     Regression

·         Linear Least Squares Fit
http://en.wikipedia.org/wiki/Regression_analysis

·         Robust Linear Regression
http://en.wikipedia.org/wiki/Robust_regression

·         Exponential Least Squares Fit
http://mathworld.wolfram.com/LeastSquaresFittingExponential.html

·         Logarithmic Least Squares Fit
http://mathworld.wolfram.com/LeastSquaresFittingLogarithmic.html

·         Power Least Squares Fit
http://mathworld.wolfram.com/LeastSquaresFittingPowerLaw.html

·         Invers Least Squares Fit
http://mathworld.wolfram.com/LeastSquaresFitting.html

·         Invers Least Squares Fit2
http://mathworld.wolfram.com/LeastSquaresFitting.html

·         n-dim. Polynomial
http://mathworld.wolfram.com/LeastSquaresFittingPolynomial.html

·         Quadratic Polynomial
http://en.wikipedia.org/wiki/Quadratic_function

·         Cubic Polynomial
http://en.wikipedia.org/wiki/Cubic_function

·         Sine Wave

·         Line Form

·         Circle Form

·         Ellipse Form

·         Inner Circle Arcs

1.3     Sample Consensus

·         Random Sample Consensus (RANSAC)
http://en.wikipedia.org/wiki/RANSAC

·         Progressive Sample Consensus (PROSAC)

·         Randomized RANSAC (RRANSAC)

·         Least Median of Squares (LMEDS)

·         M-Estimator Sample Consensus (MSAC)

·         Randomized MSAC (RMSAC)

·         Maximum LikeLihood Estimation Sample Consensus (MLESAC)

1.4     Approximation

·         Standard Simplex Algorithm

·         Gauß Algorithm

·         Bezier

·         Bezier V2

·         B-Spline

·         Smoothing Spline

·         Param. Smoothing Spline

·         Cyclic Smoothing Spline

 1.4.1          Simplex-Fit

Der Simplex-Algorithmus ist ein sehr effektives Verfahren für einen nichtlinearen Fit. Die Simplex-Algorithmus führt eine nichtlineare Näherung aller Parameter aus, die in der zu markierenden Formel markiert wurden. Wie immer in nichtlinearer Näherung sind gute Startschätzungen (für die nichtlinearen Parameter) unerlässlich.. 

1.4.2          Gauß-Fit

Der Gauß-Fit geht von der gleichen Problemdefinition aus wie der Simplex-Fit.

1.4.3          Bézier-Curves

1.4.4          Smoothing Spline

1.4.5          Parametric Smoothing Splines

1.4.6          Cyclic Smoothing Splines

1.5     Interpolation

SimplexNumerica verfügt über die folgenden Interpolatios-Algorithmen:

·         Polygonal Segments

·         Forwardflat Segments

·         Backwardflat Segments

·         Additional Segments

·         Cubic Spline

·         Parametric Spline

·         Periodic Spline

·         Cyclic Spline

·         Smooth Spline

·         Akima Subspline

·         Renner Subspline

·         Polynomial Interpolation

·         (n-1) Polynomial

·         Lagrange

·         Newton

·         Rationale

·         Aitken/Neville

·         Hermite Splines

·         Catmull-Rom Spline

·         Kochanek-Bartel Spline

·         Cardinal Spline

·         Modified Cubic Splines

·         Natural Overshooting

·         Non-overshooting in 1st Derivative

·         Non-overshooting in 2nd Derivative

·         Parabolic Interpolation

·         Fritsch-Butland Interpolation

·         Akima Interpolation

·         Kruger Interpolation

·         Extrapolation Cubic Spline

1.5.1          Polygonal Curve

1.5.2          Additive Segmentation

1.6     2D Surface Interpolation & Approximation

2D Surface Interpolation

·         Bi-Linear

·         Nearest Neighbors Linear

·         Smoothing Spline

·         Thin Plate Surface Spline

2D Surface Approximation

·         Nearest Neighbors Distance

·         Nearest Neighbors Around Distance

·         Thin Plate Surface Spline

·         Bivariate Cubic Spline

• Thin Plate Surface Spline Interpolation
• Thin Plate Surface Spline Approximation
• Bivariate Cubic Spline Approximation
• Nearest Neighbours Distance Weighted
• Nearest Neighbours Linear Interpolation
• Nearest Neighbours Around Distance Weighting

 

1.6.1          Thin Plate Surface Spline

Thin Plate Surface Spline oder kurz TPS ist eine Interpolationsmethode, bei der eine "minimal gebogene" glatte Oberfläche gefunden wird, die alle angegebenen Punkte durchläuft. TPS von 3 Kontrollpunkten ist eine Ebene, mehr als 3 ist im Allgemeinen eine gekrümmte Oberfläche und weniger als 3 ist undefiniert.

 

1.6.2          Bivariate Cubic Spline

1.6.2.1                       Nearest Neighbor Algorithm

 

1.7     Digital Signal Processing

1.7.1          Simple Waveforms

§  Sine Waveform

§  Sine Waveform with Harmonics

§  Square Waveform

§  Triangle Waveform

§  Sawtooth Waveform

§  Gaussian Noise

1.7.2          Complex Waveforms

§  Summation of sine waves

§  Sawtooth wave by summation

§  Sawtooth wave by direct calculation

§  Inverse sawtooth wave

§  Triangle wave

§  Square waves

§  Pulse wave                                                                     

§  Frequency modulation FM

§  Phase modulation PM

§  Amplitude modulation AM

§  Ring modulation AM                                                                  

§  Pulse wave buzz

§  Pulse wave buzz2                                                                         

§  Dynamic spectrum

§  Bandwidth limited

§  Waveshaping

§  White noise

1.7.3          Envelope Generators                    

o   Simple Linear Integration

o   Convex Exponential Interpolation

o   Variable Exponential Interpolation

o   Logarithmic Interpolation

o   DB Interpolation

o   Simple State Machine

o   Multiple Segments ADSR

o   Multiple Segments State Machine                                                        

o   Constant Rate ADSR

o   Constant Rate ADSR Transformed                                                        

o   Amplitude Envelope                                   

1.8     DFT 

·         Approximation

·         Spectrum

·         Phase

·         Real Part

·         Imaginary Part

1.9     FFT 

·         Approximation

·         Spectrum (Simplex FFT)

·         Spectrum (Complex FFT

·         Spectrum (Real FFT)

·         Phase

·         Real Fourier Analysis

·         Real Fourier Synthesis

·         Real Part

·         Imaginary Part

·         Interpolation (2:1 ratio)

 

1.9.1          Fast Fourier Transform (FFT)

1.9.1.1                       Fourier Subtotal

 1.9.1.2                       Spectrum

 1.9.1.3                       Phase Angle

1.9.1.4                       Fourier Analysis

 1.9.1.5                       Fourier Synthesis

1.9.1.6                       Imaginary part

1.10    Fenster-Funktionen

·         None Window (Rectangle)

·         Hamming Window

·         Hanning Window

·         Hann Window (Matlab)

·         Blackman Window

·         Triangular Window

·         Sqrt Window

·         Hann Window (Numerical Recipes)

·         Blackman-Harris Window

·         Bartlett Window

·         Parzen Window

·         Welch Window

·         Steeper Window

·         Kaiser Window

·         Flat Top Window

·         Flat Top Window (Stanford)     

1.11    Goertzel DFT (Filter)   

·         Scaled Magnitude

·         Tone detection

               

1.12    FIR Filter      

·         FIR Coefficients (Remez Algorithm)

·         FIR Filter Graph (Remez Algorithm)                                      

·         FIR Coefficients (Parks McClellan Algorithm)

·         FIR Filter Graph (Parks McClellan Algorithm)                                    

·         FIR Coefficients (Iowa Hills Algorithm)

·         FIR Filter Graph (Iowa Hills Algorithm)

               

1.13    IIR Filter

·         RBJ Biquad

·         Butterworth

·         Chebyshev I

·         Chebyshev II

·         Elliptic

·         Bessel

·         Legendre

·         Custom                                             

               

1.14    Zeitreihen Filter  

·         Simple Moving Averages

·         Exponential Moving Averages

·         Linear Moving Averages                                            

·         Savitzky-Golay Filter

·         FIR / IIR Real Filter

               

1.15    Audio Signale

·         Play                                     

·         Change Sound Tempo

·         Change Sound Pitch

·         Change Sound Rate

·         Adjust Tempo to BPM

               

1.16    Mathematische Operationen

·         Add Array

·         Add Number

·         Sub Array

·         Sub Number

·         Mul Array

·         Mul Number

·         Div Array

·         Div Number

1.17    Mathematische Funktionen

·         Formula Parser

·         Bernstein Polynomial [0,1]

·         Bernstein Polynomial [a,b]

·         Bernoulli Polynom of order N

·         Euler Beta Function (Euler's Integral)

·         Gaus Error Function

·         Gaus Error Function Inverse

·         Legendre Polynomial

·         Hermite Polynomial

·         POPUP Bessel Functions

·         Bessel Function of the first kind

·         Bessel Function of the second kind

·         Modified Bessel Function of the first kind

·         Modified Bessel Function of the second kind

·         Spherical Bessel Function of the first kind

·         Spherical Bessel Function of the second kind

 

1.17.1     Airy Funktionen

·         Airy function Ai

·         Airy function Bi

·         Airy function Ai'

·         Airy function Bi'

 

1.17.2     Elliptische Integrale

·         Carlson's elliptic integral RC

·         Legendre's complete elliptic integral of the first kind

·         Legendre's complete elliptic integral of the second kind

·         Legendre's complete elliptic integral of the third kind

 

1.17.3     Jacobi Elliptische Funktionen

·         Function cd

·         Function cn

·         Function cs

·         Function dc

·         Function dn

·         Function ds

·         Function nc

·         Function nd

·         Function ns

·         Function sc

·         Function sd

·         Function sn

·         Riemann Zeta Function (left)

·         Riemann Zeta Function (right)

1.18    Differentiation

1.19    Integration

1.20    Ausreißer Algorithmen

·         Auto Detection

·         Dean-Dixon Outlier Test

·         Nalimov Outlier Test

·         Grubbs Outlier Test

·         Significance of extreme values

·         Show Outlier Test Limit

·         Show Outliers in Output Window

·         Acoustics Alarm if any Outlier

1.21    Daten Reduktion

·         Routine from MIR (Russian Space Station)

·         Band Slope Method

·         Scherenschnitt Method

·         Removing nth Data Points

·         Radial Vertex Reduction

·         Perpendicular Vertex Reduction

·         Retake Perpendicular Vertex Reduction

·         Reumann/Witkam Reduction

·         Ramer/Douglas/Peucker Reduction

·         Optimized Ramer/Douglas/Peucker Reduction

·         Opheim Simplification

·         Lang Simplification

1.22    Konvexe Hüllen

·         Hull Edge Points

·         Hull Polygon

·         Hull Curve

1.23    Histogramme

·         Histogramme

·         Inter Line

·         Stochastic Alpha, BetaRho (SABR)

1.24   Peak Detektion / Peak Finder

·         Emission Peak Finder

·         Absorption Peak Finder