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math and numerical models |
In this lecture, we introduce for numerical analysis by comparing between analytical and numerical models |
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matlab (handling scalar variables) |
Introduction to matlab environment and handling scalar variables and several basic operations on them |
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2. |
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matlab (handling vector variables) |
handling vector variables in matlab and several basic operations on them including functions and plotting |
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3. |
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matlab (handling matrices variables) |
handling vector variables in matlab and several basic operations on them including functions and plotting |
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4. |
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programming with matlab |
introduction to more advanced topic in matlab such as plotting, inline functions, handling complex equations etc. |
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5. |
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Roundoff and Truncation Errors : Matlab hints |
analyze sources of errors in computations with focus on reducing the errors |
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Roots: Bracketing methods : Introduction & Bracketing methods |
introducing bracketing methods for finding the intervals of roots |
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Roots: Bracketing methods : open methods & matlab hints |
introducing open methods for finding the intervals of roots |
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Linear algebraic equations : Introduction to matrices |
introduce simple linear systems and several special matrices and their use |
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Linear algebraic equations : matrices operaions |
Introduce several simple matrix-matrix and matrix-vector operations |
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Linear algebraic equations : singular value decomposition |
Shed light o singular valus and their use in analyzing the stability of systems |
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7. |
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Gauss elimination: Cramer's rule & Naive Gauss elimination |
Introduce Cramer's rule and Gausse elimiation as methods to solve linear algebraic systems |
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LU decomposition and Cholesky decomposition: Gauss elimination |
Introduce LU and Cholesky decompositoon methods and introdce for matrix inversion |
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8. |
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Matrix inversion using LU decomposition |
Due to its low complexity and stability, LU is used as a matrix inversion technique |
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Error analysis and system condition |
In presence of noise, the system performance is analyzed using already studied algorithms including matrix inversion. Also, the idea and implementation of the maximum likelihood are investigated |
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9. |
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Iterative method - Gauss-seidel method |
introduce iterative methods to solve linear systems - Gauss-Seidel method |
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Iterative method - newton-raphson method |
introduce nonlinear systems and introduce the newton-raphson method |
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Introduce EVD & SVD |
Introduction to singular value decomposition and eigen value decomposition |
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EVD (eigen value decompositon) |
Introduce the characteristic polynomial and how to obtain the eigen values and relation with trace and det of a matrix |
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Power method |
Introduce the power method to numerically obtain the minimum eigenvalue of a matrix |
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EVD vs. SVD |
introduce points of similarity and difference between eigenvalues and singular values |
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10. |
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Introduction & overview |
introduce curve fitting |
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Liner least-squares regression |
introduce the linear least squares regression and the derivation of the line parameters |
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Linearization of nonliner models |
linearization of power, exponential and saturation rate nonlinear models |
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Practice explanation |
A practice on the linear curve fitting and the linearization of nonlinear models |
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11. |
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introduction (and reply to twitter questions) |
more examples of linear curve fitting and linearization of nonlinear models |
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polynomial regression - single variable |
introduce the polynomial regression (nonlinearregression) and the derivation of the polynomial parameters |
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polynomial regression - 2 variables |
apply the nonlinear regression to the case of two independent variables |
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12. |
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Introduction (~ slide 6) |
Introduction to interpolation |
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Newton interpolation polynomial |
introduce the newton interpolation polynomial and discuss accuracy of this method |
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Lagrange interpolation polynomial |
introduce the lagrange interpolation polynomial and discuss accuracy of this method |
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Extrapolation |
introduce the idea behind extrapolation and the danger of this technique specially when using high order polynomials |
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13. |
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Integration and Newton-Cotes Formulas |
Introduce the basic idea of numerical integration and the newton-cotes method that replaces the function with an interpolation function over which the integration is performed |
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Integration: The Trapezoidal/composite Trapezoidal rule |
Introduce the Trapezoidal method which consists of integration over a line and a method of improving the result via several integration over several several intervals |
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Simpson's rules |
Introduce the simpson's 1/3 and 3/8 rules which consist of integration over second order and third order polynomials, respectively |
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Richardson Extrapolation |
Introduce the Richardson method which consists of using two less accurate integrals to obtain a more accurate result |
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