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Numerical-Computing-slides-chapter1 - Numerical Computing...

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Numerical Computing– Chapter 1 Dr. Nguyen V.M. Man Email: [email protected] Faculty of Computer Science and Engineering HCMUT Vietnam February 23, 2010
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Abstract This talks presents selective topics of Numerical Computing from theoretic views to practical applications at HCMUT- Spring 2009
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The structure of the course ——————————————————————————— Part I: Approximation of polynomials Part II: Direct and iterative methods for linear systems Part III: Eigenvalues problems Part IV: Numerically solving of non-linear system Part V: Numerical solutions of ordinary differential equations
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Part I: Approximation of polynomials ——————————————————————————— Polynomial interpolation - Piecewise polynomial interpolation Hermite interpolation - Interpolation in two dimensions Least-squares methods References X An Introduction to Numerical Methods and Analysis, by James F. Epperson John Wiley & Sons, 2002 X Other softcopys
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Polynomial interpolation - Piecewise polynomial interpolation ——————————————————————————— What is approximation? In general, approximation means the representation of arbitrary objects or phenomena by other simpler objects s.t. the main characteristics are preserved A continuous phenomenon occurring in reality is to be described using continuous mathematical models and why it is important ? Information (in numerical methods sense): defined to be what is known about an object or a system Discrete information Algebraic data; Continuous one Analytic data Easy to obtain algebraic data!
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Structural models ——————————————————————————— Examples. Structural models : as neural networks, the mathematical model of small set of human nerve cells. The model keeps the structure of real system of nerve cells as well as the way in which they transmit information.
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and Functional models ——————————————————————————— Functional models . Air-pollution prediction. We can identify air-pollution prediction with a time-series recursive model describing the atmospheric concentration of SO 2 . Let Y [ k + 1] be the atmospheric concentration of SO 2 at time point k + 1 in HCMC, then: Y [ k + 1] = a · Y [ k ] + b ( T [ k + 1] + c ) 2 + d V [ k + 1] + e , where I a , b , c , d , e : scalars, I T [ k ]: predicted average temp., and I V [ k ]: predicted average wind speed in the k -th day.
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Example 1: SO 2 prediction ——————————————————————————— Y [ k + 1] = a · Y [ k ] + b ( T [ k + 1] + c ) 2 + d V [ k + 1] + e , The past average SO 2 concentration values measured every hour are discrete information. We can obtain algebraic data by creating vectors of measured values Discrete and continuous information . The aim of mathematical modeling and approximation: obtain analytic data which deviates from the original phenomenon only within given bounds.
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