SimplexNumerica - Algorithm Manual.pdf - Opisy programów 2 - mars2323

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SimplexNumerica

Simplex

Numerica

Algorithm Manual

DRAFT A

Version 8.0

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SimplexNumerica

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SimplexNumerica

THE SIMPLEXNUMERICA AND SIMPLEXPARSER PROGRAMS ARE PROVIDED "AS IS"

WITHOUT WARRANTY OF ANY KIND, EITHER EXPRESS OR IMPLIED, INCLUDING, BUT NOT

LIMITED TO WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR

PURPOSE. IN NO EVENT WILL THE AUTHOR BE LIABLE TO YOU FOR ANY DAMAGES,

INCLUDING INCIDENTAL OR CONSEQUENTIAL DAMAGES, ARISING OUT OF THE USE OF THE

PROGRAM, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGES.

Dipl.-Phys.-Ing. Ralf Wirtz

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Contents

SimplexNumerica.....................................................................................................................................3

Contents ....................................................................................................................................................4

1 2D Algorithm ....................................................................................................................................5

1.1 Regression ..............................................................................................................................................5

1.1.1 Linear Least Squares...........................................................................................................................................6

1.1.2 Polynomial Regression .....................................................................................................................................10

1.1.3 Linearization of Nonlinear Relationships .........................................................................................................12

1.2 Interpolation ........................................................................................................................................14

1.2.1 Polygonal Curve ...............................................................................................................................................15

1.2.2 Additive Segmentation .....................................................................................................................................16

1.2.3 (n-1)-Polynomal................................................................................................................................................17

1.2.4 Lagrange Interpolation......................................................................................................................................17

1.2.5 Newton Interpolation ........................................................................................................................................18

1.3 Spline Interpolation.............................................................................................................................20

1.3.1 Cubic Splines ....................................................................................................................................................20

1.4 Approximation .....................................................................................................................................23

1.4.1 Gauss/Cholesky ................................................................................................................................................23

1.4.2 Simplex-Fit .......................................................................................................................................................27

1.4.3 Gauß-Fit............................................................................................................................................................34

1.5 Fourier Analysis of Time Series .........................................................................................................37

1.5.1 Discrete Fourier Transform (DFT) ...................................................................................................................37

1.5.2 Fast Fourier Transform (FFT)...........................................................................................................................39

1.6 Bezier & Spline Approximation .........................................................................................................49

1.6.1 Bézier-Curves ...................................................................................................................................................49

1.6.2 B-Spline ............................................................................................................................................................56

1.6.3 Smoothing Spline..............................................................................................................................................58

1.6.4 Parametric Smoothing Splines ..........................................................................................................................63

1.6.5 Cyclic Smoothing Splines.................................................................................................................................66

1.7 Numerical Integration.........................................................................................................................67

1.7.1 Rectangular Rule...............................................................................................................................................67

1.7.2 Error Bounds and Estimate for Rectangular Rule.............................................................................................68

1.7.3 Trapezoidal Rule...............................................................................................................................................68

1.7.4 Error Bounds and Estimate for Trapezoidal Rule .............................................................................................69

1.7.5 Simpson’s Rule .................................................................................................................................................71

1.7.6 User-Interface ...................................................................................................................................................74

1.8 Numerical Differentiation ...................................................................................................................77

1.9 Statistics................................................................................................................................................79

1.9.1 Basic Probability...............................................................................................................................................79

1.9.2 Population parameters.......................................................................................................................................80

1.9.3 Mean, Variance, and Standard Deviation .........................................................................................................80

1.9.4 Covariance and Correlation ..............................................................................................................................81

1.9.5 Student’s T-Test................................................................................................................................................82

1.10 Contour Plot.........................................................................................................................................83

1.11 Convex Wrapper Polygon...................................................................................................................87

1.12 Convex Wrapper Curve (Envelope) ..................................................................................................87

1.13 Edge Points...........................................................................................................................................87

2 3D Approximation and Interpolation ............................................................................................88

2.1 Thin Plate Surface Spline....................................................................................................................88

2.2 Bivariate Cubic Spline.........................................................................................................................90

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2D Algorithm

1 2D Algorithm

Curve and surface algorithm are an important topic in SimplexNu-

merica for geometric modelling and visualization courses. In addition to manually set

Bezier and B-Splines from previous Pulldownmenu, interpolation and approximation pro-

vides another level of sophistication. In many situations such as surface re-engineering

and facial movement animation, you may specify a set of data points that describes a

desired shape (e.g., surface model) through any probing or scanning, and obtain a sur-

face that contains all data points. Interpolation is also important in computer animation.

An animator may specify a number of key camera positions and orientations (i.e., key

frames), interpolate these positions with a B-spline curve (i.e., camera path), and inter-

polate the key frames with additional frames. While interpolation can produce a

curve/surface that follows the shape of the data points, it may oscillate or wiggle its way

through every point. Approximation can overcome this problem so that the curve/surface

still captures the shape of the data points without containing all of them.

1.1 Regression

Consider the nature of most experimental data. Typically such data

include noise due to many different effects. The noisy data from an experiment might

appear as shown in the following Table and Figure. We assume that the x values are ac-

curate. Visual inspection of the data suggests a positive relationship between x and y =

f(x) , i.e., higher values of y are associated with higher values of x . One strategy for deriv-

ing an approximating function for this data might be to try to fit the general trend of the

data without necessarily matching the individual points. A straight line could be used to

generally characterize the trend in the data without passing through any particular point.

The line in next Figure has been sketched through the points. Although this approach

may work well in many cases, it does not provide us with any quantitative measure of

how good the fit of the line is to the data. We need a criterion with which to measure the

goodness of fit of the line to the data. One way to do this is to derive a curve that mini-

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