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By Edwin Zondervan

"This booklet emphasizes the deriviation and use of various numerical tools for fixing chemical engineering difficulties. The algorithms are used to resolve linear equations, nonlinear equations, usual differential equations and partial differential equations. it is usually chapters on linear- and nonlinear regression and ond optimizaiton. MATLAB is followed because the programming atmosphere in the course of the book. Read more...

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Extra resources for A numerical primer for the chemical engineer

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10 Summary In this chapter we found that linear equations can be written as matrices. Whether or not a solution exists, depends on the rank of a matrix. We also showed briefly what eigenvectors and eigenvalues are. Such matrix properties are useful in determining whether a system can be solved, or if a system is stable or not. a The following linear system is given: 2x1 + x2 + x3 x1 + 2x2 + 2x3 = = 4 3 x1 − x2 + 6x3 = 1. Rewrite this system in terms of Ax = b and then determine A−1 with use of cofactors.

5 1 0  , P = U =  0 0 0 0 4/3 0 2/3 1 1 0  0 1 . 5 Summary In this chapter we wrote a program that can solve a system of linear equations using Gaussian elimination and back substitution. This method is rather slow for large systems. MATLAB has a good solver of A\b itself. We found that back substitution is relatively fast and that repeatedly performing row operations slows down the solution process a lot. Decomposing a matrix into an L and a U matrix can be used to perform row operations systematically and much faster.

0 0 0 0 Here, rank(M ) = rank(Ma ) = 2, which is smaller than n, so there is an infinite number of solutions. 8 Eigenvalues and eigenvectors Matrices have characteristic directions, called eigenvectors. An eigenvector e is defined by: M e = λe. 27) When an eigenvector is multiplied by the matrix M , the result is the eigenvector itself. The scale constant λ is called the eigenvalue. Any multiple of an eigenvector is also an eigenvector. 27 that M e − Iλe = 0, or (M − Iλ)e = 0. In order to find values for λ, the determinant det(M − Iλ) should be zero.

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