Download PDF Engineering Analysis Interactive Methods and Programs with FORTRAN, QuickBASIC, MATLAB, and Mathematica by Y. C. Pao

Sinopsis

Computers are best suited for repetitive calculations and for organizing data into specialized forms. In this chapter, we review the matrix and vector notation and their manipulations and applications. Vector is a one-dimensional array of numbers and/or characters arranged as a single column. The number of rows is called the order of that vector. Matrix is an extension of vector when a set of numbers and/or  characters are arranged in rectangular form. If it has M rows and N column, this matrix then is said to be of order M by N. When M = N, then we say this square matrix is of order N (or M). It is obvious that vector is a special case of matrix when there is only one column. Consequently, a vector is referred to as a column matrix as opposed to the row matrix which has only one row. Braces are conventionally used to indicate a vector such as {V} and brackets are for a matrix such as [M]. In writing a computer program, DIMENSION or DIM statements are necessary to declare that a certain variable is a vector or a matrix. Such statements instruct the computer to assign multiple memory spaces for keeping the values of that vector or matrix. When we deal with a large number of different entities in a group, it is better to arrange these entities in vector or matrix form and refer to a particular entity by specifying where it is located in that group by pointing to the row (and column) number(s). Such as in the case of having 100 numbers represented by the variable names A, B, …, or by A(1) through A(100), the former requires 100 different characters or combinations of characters and the latter certainly has the advantage of having only one name. The A(1) through A(100) arrangement is to adopt a vector; these numbers can also be arranged in a matrix of 10 rows and 10 columns, or 20 rows and five columns depending on the characteristics of these numbers. In the cases of collecting the engineering data from tests of 20 samples during five different days, then arranging these 100 data into a matrix of 20 rows and five columns will be better than of 10 rows and 10 columns because each column contains the data collected during a particular day.

Content

1. Matrix Algebra and Solution of Matrix Equations
2. Exact, Least-Squares, and Spline Curve-Fits
3. Roots of Polynomial and Transcendental Equations
4. Finite Differences, Interpolation, and Numerical Differentiation
5. Numerical Integration and Program Volume
6. Ordinary Differential Equations — Initial and Boundary Value Problems
7. Eigenvalue and Eigenvector Problems
8. Partial Differential Equations

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