By Claude Brezinski

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**Extra info for Biorthogonality and its applications to numerical analysis**

**Example text**

The n × m matrix is filled in column order. Thus, rows become columns, and columns become rows. To use the data in its original order, transpose the 26 ◾ Numerical and Analytical Methods with MATLAB read-in matrix. To transpose a matrix A in MATLAB, simply type in A'. This changes columns to rows and rows to columns. ◾◾ An existing data file can also be entered into a program by the load command. The load command, unlike the fscanf command, leaves rows as rows and columns as columns. txt x = filename(:,1); y = filename(:,2); The input file must have the same number of rows in each column.

M. m by typing which factorial in the Command window. 4) % Define a function, 'exf1', that evaluates e^x. % This function takes 'x' as an input argument and is % called either from the Command Window or from another % program. The resulting output, 'ex', is available to % be used in another program or in the Command Window. % In this example, term(n+1) is obtained from term(n) % by multiplying term(n) by x/n. % Note: e^x = 1 + x + x^2/2! + x^3/3! + x^4/4! + ... 0e-7 break; end end ex=sum(term); ------------------------------------------------------------ To test out this function, run it from the Command window.

For example, mod(13,5) = 3, that is, 13 ÷ 5 gives 2 plus remainder of 3 (the 2 is discarded). As another example, mod(n,2) gives zero if n is an even integer and one if n is an odd integer. Examples (try typing these statements into the Command window): clc; A = [2 15 6 18]; length(A) y = max(A) z = sum(A) A = [2 15 6 18; 15 10 8 4; 10 6 12 3]; x = max(A) y = sum(A) size(A) mod(21,2) mod(20,2) ◾◾ A list of the complete set of elementary math functions can be obtained by typing help elfun in the Command window.