# 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)

·         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

·         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.

## 1.5     Interpolation

SimplexNumerica verfügt über die folgenden Interpolatios-Algorithmen:

·         Polygonal Segments

·         Forwardflat Segments

·         Backwardflat 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

## 1.6     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.

## 1.7     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 State Machine

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

## 1.10    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

·         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

## 1.16    Mathematische Operationen

·         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)

## 1.20    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

·         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