Scientific Computing 2020-21

Attention - how the course would look like

Office hours:

Homeworks - lab problems

References

1. Piotr Krzyşanowski, Obliczenia in�ynierskie i naukowe. Skuteczne, szybkie, efektowne, Wydawnictwo Naukowe PWN, 2011
2. Piotr KrzyĹźanowski, Obliczenia naukowe. Html lecture notes (in Polish). Pdf version is also available there. University of Warsaw. 2010.

Lab

• Lab LINUX BASICS AND OCTAVE :
  • linux basics: cd, ls, rm, cp, pwd, mv, tar, mkdir, rmdir, chmod, head, tail, more, cat, man.
  • Basics of octave - octave as a scientific calculator, semicolon : how to create a matrix, saving/loading from/to files in different formats; basic matrix/vector operations - opt. solving linear equations and LLSP etc
  • Sone simple problems
    1. Create matrices 3x4 A and 3x5 B - then the matrix 3x8 C with 3 first columns are A and next B. Create D 3x3 as a main minor of C oi.e. from C(1,1) to C(3,3). Change the order of columns of D. Compute sin on elements of D. Write D to a file (binary or ascii) - change D(1,1) to -100 and load this new matrix to octave as DD. Compute 1st norm of DD.
    2. Compute a discrete max norm of (sin(x))^2 on [0,1] (on 10000 points)
    3. Create a text file with any 3x3 matrix in a proper octave format and load it to octave.
    4. Plot a graph of sin(x) on [0,4]. Do the same for x*(sin(x)+2).
    5. Compute an approximation of the derivative by central difference on [0,4] on N+1 equidistant points with h=4/N N=100 i.e. compute Dcf(xk)=0.5(f(xk+h)-f(xk-h))/h for xk=4*k*h k=1,..,N (equidistant points on [0,4]. Can it be done without a loop? Compare with the values of analytical derivative at those points i.e. er(xk;h)=|Dcf(xk)-f'(xk)| compute Er(h)=max_k(er(xk;h)). Check the order of approximation i.e. take once N=100 then N=200 and compute Er(h)/Er(h/2) or plot er(xk;h)/er(xk;h/2). Repeat the problem for forward and backward differences respectively (fwd diff: (f(xk+h)-f(xk))/h; bck diff (f(xk)-f(xk-h))/h)
    6. Compute an approximation of the 2nd order derivative by 3point difference on [0,4] on N+1 equidistant points with h=4/N N=100 i.e. compute D2f(xk)=(f(xk+h)-2f(xk)+f(xk-h))/(h*h) for xk=4*k*h k=1,..,N (equidistant points on [0,4]. Compare with the values of analytical 2nd derivative at those points i.e. er(xk;h)=|D2f(xk)-f''(xk)| compute Er(h)=max_k(er(xk;h)). Check the order of approximation.
  
  Lab - scripts: example
  matbasic.m write on command line of octave'a: matbasic.
  lab1.m some solutions
  matB.txt an xample of a saved matrix (can be loaded into an octatave session)
• Lab - functions, m-files, basic linear algebra, solving the systems, LLSP, norms, matrix factorizations: Cholesky, LU, QR, eigenvalue problems etc
  • Write a m-file with a function which has two parameters x and a and returns f(x;a)=x^a and its derivative as a 2nd returned value; default value of a is one.
  • Solve any linear 2x2 system e.g. A=[1,2;3,4] f=[1;3], x=A\f - test if residual vector f-A*x is zero, and if we get the right solution (if we know it) here the solution =[1;0]
  • Solve any simple regular LLSP - e.g. A=[1,1;1,2;3,4] f- the 1st column of A - x=A\f - is the computed solution equal to the 1st unit vector ie. to [1;0]?
  • A simple problem - polynomial interpolation; and linear regression as a LLSP
    • using vander() and polyval() and backslash - solve linear regression problem for givem x_k,y_k find a,b such that \sum_k |ax_k+b -y_k|^2 is minimal. Compare with polyfit(). Write a function which for two parametes being the vecors x and y (of the same length) returns the vector wit hthe parameters [a;b] of the linear regression problem.
      More demanding version : The function should check if the problem has a unique solution (up to fl calculations naturally)? How to do it withou using rank(), svd(), or null() functions? HINT: perhaps we can use qr() instead of backslash?
    • find Lagrange interpolating polynomial on equidistant points on [-5,5] for f(x)=1/(1+x*x) - compute discrete max norm of the error on [-5,5] and at interpolation nodes Test for 5,10,20 and 30 nodes.
    • do same as in the previous problem but for Chebyshev points (linearly scaled roots of T_n(x)=cos (n*arc cos(x))).
    • Not always direct solvers gives a good solution. Take N=2+5*k equidistant nodes on [0,1] and repeat computing Lagrange interpolator p for any function e.g. f(x)=sin(3x) for k=0,1,2,3,...; check if f=p at interpolation nodes i.e compute max_k|f(x(k)-p(x(k))| where (x(k))_k interpolation nodes.
  • Take a symmetric matrix 2x2 and compute its eigenvalues and eigenvectors. Check if AX=XD X- a matrix having the eigenvectors as columns - D a diagonal matrix with with respective eigenvalues on its diagonal. Test for A=[2,1;1,2], then D=diag([3;1]) and X=[1,1;1,-1]. Compare with the results of eig(). Repeat tests for different random symmetric matrices.
  • SVD: compute svd() for a random matrices Check if the obtained values are OK.
  • Solve a LLSP problem by svd() for a random matrix 5x4 and 4x5. Compare with backslash results.
  • Compute using svd() condition of A for a random A 5x5; compute the second matrix norm of A for a random A 5x4 etc Compute the orhonormal basis of ker(A) and R(A). Compare with the results obtained by null() and orth().
  
  

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  A few octave scripts, m-files, source C files, LAPACK functions

  Link to Octave
  

  Octave scripts, m-files,simple C files with e.g. examples of an use of BLAS/LAPACK

  matbasic.m - a script with basic octave commands we can edit it : gedit matbasic.m

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  Wyklad - srodowko programisty gdzie mozna znalezc materiały o jezyku C jak i Makefile'ach (in Polish)

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  Project

  Later in April - May 2021

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  Return to my homepage
  
  

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  Last update : March 3, 2021
  
  

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