Edwin Zondervan's A numerical primer for the chemical engineer PDF

By Edwin Zondervan

ISBN-10: 1482229447

ISBN-13: 9781482229448

"This publication emphasizes the deriviation and use of numerous numerical tools for fixing chemical engineering difficulties. The algorithms are used to unravel linear equations, nonlinear equations, usual differential equations and partial differential equations. it's also chapters on linear- and nonlinear regression and ond optimizaiton. MATLAB is followed because the programming setting through the book. Read more...

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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. The L and U matrices can directly be solved using forward and back substitution.

028. The relative error increases strongly! x˜ and y˜ both have 5 significant digits, but the difference x ˜ − y˜ only has 2 significant digits: by performing the subtraction, significant digits are lost! 14. 006738. But, if you only use numbers fixed on 4 digits for your calculation, you can observe the results in the following table: n 0 1 2 3 4 5 6 ··· 23 24 25 1 + (−5)0 /1! + ... + (−5)n /n! 009989, something remarkably different as the real value. The continuously changing positive and negative sign is the cause of the loss of digits in this problem.

1  .  =  e1 e2 e3   0 e e e M 1 2 3     .. .. .. 0 . . . 34) or more compactly written as M U = U Λ ↔ M = U ΛU −1 . 35) We will use this spectral decomposition later on. , det(A − λI) = 0 has repeated roots. 36) where Λ = diag(λ1 , λ2 , · · · , λN ). Proof : We first write A as a Schur decomposition: A = U RU H . 38) we then form the two matrix products: AAH = U RU H (U RH U H ) = U RRH U H AH A = U RH U H (U RU H ) = U RH RU H . 40) 22 A Numerical Primer for the Chemical Engineer For A to be normal, AAH must be normal  R11 R12 R13  R22 R23   R33 R=   and RH     =   ¯ 11 R ¯ 12 R ¯ 13 R ..

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A numerical primer for the chemical engineer by Edwin Zondervan

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