Elements of numerical linear algebra and quadrature

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Emma Perracchione

Corso di Calcolo Numerico per Ingegneria Meccanica - Matr. PARI (Univ. PD)

Gli esercizi sono presi dal libro: S. De Marchi, D. Poggiali, Exercices of numerical calculus with solutions in Matlab/Octave.

A.A. 2018/2019

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Elements of numerical linear algebra and quadrature Materiale

Materiale

TUTTO IL MATERIALE SI TROVA AL SEGUENTE LINK E VERRA’

AGGIORNATO AD OGNI LEZIONE.

https://www.math.unipd.it/~emma/CN1819.html

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Elements of numerical linear algebra and quadrature Exercises

Exercise 1

Exercise

On a script called Esercizio 1 compare SimpsonComposto and TrapezioComposto for the following two integrals with the same number of nodes (n = 11):

Z ₁

0

1 + x² dx . Z 1

0

1/8 + 2x dx . Compute the error.

Comment the results.

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Elements of numerical linear algebra and quadrature Solving systems

Matlab routines

For solving a system Ax = b, with A ∈ R^n×n,x, b ∈ Rⁿ in Matlab, we simply have to write x=A \ b.

\ Backslash or left matrix divide.

A\B is the matrix division of A into B, which is roughly the same as INV(A)*B , except it is computed in a different way.

If A is an N-by-N matrix and B is a column vector with N components, or a matrix with several such columns, then X = A\B is the solution to the equation A*X = B. A warning message is printed if A is badly scaled or nearly singular.

A\EYE(SIZE(A)) produces the inverse of A.

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Elements of numerical linear algebra and quadrature Remarks

Gauss

Given a square matrix A ∈ R^n×n of elements a_ik, i , k = 1, . . . , n, for performing the gaussian reduction, we need to calculate for

k = 1, . . . , n − 1

m_ik = a^(k)_ik a^(k)_kk

, i = k + 1, . . . , n,

and then

a^(k+1)_ij = a^(k)_ij − m_ika^(k+1)_kj , i , j = k + 1, . . . , n.

One can interpret this as a factorization of the type LU = A, where the entries of L are given by m_ik (and a diagonal of ones) and the

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Elements of numerical linear algebra and quadrature Exercices

Exercise 2

Exercise

Write the function for LU factorisation without pivoting (and call it LUnoPiv).