## Lecture: Least-Squares Fitting Methods

The basic theory of curve fitting and least-square error is developed....

From AMATH 301

The basic theory of curve fitting and least-square error is developed....

From AMATH 301

Polynomial fitting of the data, via Lagrange polynomials, can also be considered as the fit curves go through all data points. Spline technology is developed to circumvent polynomial wiggle....

From AMATH 301

We develop a MATLAB code that implements all the theoretical methods considered for curve fitting: least-square fits, polynomial fits and splines....

From AMATH 301

We introduce one of the most fundamental concepts of linear algebra: eigenvalues and eigenvectors...

From AMATH 301

We introduce some of the basic techniques of optimization that do not require derivative information from the function being optimized, including golden section search and successive parabolic interpolation....

From AMATH 301

We develop a theoretical approach to understanding how eigen-decompositions of matrices can be used in iterative schemes for Ax=b....

From AMATH 301

This details how to apply a simple iteration procedure for solving Ax=b, including Jacobi iterations and Gauss-Siedel modifications....

From AMATH 301

This lecture details the algorithm used for constructing the FFT and DFT representations using efficient computation....

From AMATH 301

Derivative-based methods are some of the work-horse algorithms of modern optimization, including gradient descent....

From AMATH 301

Outline of the basic theory of the Fourier Transform and the representation of data in the frequency domain...

From AMATH 301

We consider a number of more advanced optimization algorithms that include the genetic algorithm and linear programming for constrained optimization....

From AMATH 301

Using the definition of derivative and Taylor series, numerical time-stepping schemes are produced for predicting the future state of ODE systems....

From AMATH 301

From simple Taylor series expansions, the theory of numerical differentiation is developed....

From AMATH 301

Higher-order numerical integration schemes are considered along the classic schemes of trapezoidal rule and Simpsonâ€™s rule....

From AMATH 301

The applications of the FFT are immense. Here it is shown to be useful in compressing images in the frequency domain....

From AMATH 301

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From AMATH 301

The accuracy of the differentiation approximations is considered and new schemes are developed to lower the error. Integration is also introduced as a numerical algorithm....

From AMATH 301

A general framework for time-stepping schemes is developed, culminating in the 4th-order accurate Runge-Kutta scheme which is the work-horse of many applications....

From AMATH 301

The accuracy and stability of time-stepping schemes are considered and compared on various time-stepping algorithms....

From AMATH 301

We finish by considering the physical application of a double pendulum and a numerical model for its motion, demonstrating the chaotic behavior induced in the motion....

From AMATH 301

We show how ODE time-steppers can be used to study myriad of dynamical trajectories simultaneously in a vectorized form....

From AMATH 301

We demonstrate the application of the 4th-order accurate Runge-Kutta solver (ODE45) to the classic Lorenz system....

From AMATH 301

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From AMATH 301

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From AMATH 301

Perhaps the most important concept in this course, an introduction to the SVD is given and its mathematical foundations....

From AMATH 301

The SVD algorithm is used to produce the dominant correlated mode structures in a data matrix....

From AMATH 301

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From AMATH 301

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From AMATH 301

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From AMATH 301

We demonstrate the power of the SVD/PCA framework on the computer vision problem of face recognition...

From AMATH 301