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Computing Techniques

by: Chloe Reilly

Computing Techniques ME 2016

Chloe Reilly

GPA 3.62

Marc Smith

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Marc Smith
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This 0 page Class Notes was uploaded by Chloe Reilly on Monday November 2, 2015. The Class Notes belongs to ME 2016 at Georgia Institute of Technology - Main Campus taught by Marc Smith in Fall. Since its upload, it has received 8 views. For similar materials see /class/234243/me-2016-georgia-institute-of-technology-main-campus in Mechanical Engineering at Georgia Institute of Technology - Main Campus.

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Date Created: 11/02/15
GUASSIAN ELIMINATION WITH PIVOTING If the diagonal element the pivot element of a row is zero the algorithm will divide by zero To prevent this switch this row with one below it Decide which row to use based on the element with the largest absolute value This rule also reduces roundoff error in the computation MATLAB CODE function L U piv GEpivA Gaussian elimination with pivoting input A a square matrix output L and U the LU factors of A piv the pivoting vector of A m n sizeA should check to insure that n m piv ln de ne pivoting vector for k lnl pivot rows if necessary maxr r maX absAknk q r k l ink q inq kl Alk Cl 1 A61 k I Page 1 MATRIX MULTIPLICATION C AB Def Ci ZAjkBk k1 The matrix sizes are A is m X n B is n X p C is m X p Sum the elements multiplied by the elements of this row of this column A11 A12 39 39 39 Aln Bll Blj Blp BZI BZj 3217 C 2 A71 AiZ Am i I Aml AmZ I I I A mn Bnl an Bnp The inner dimension n of A and B must be the same MATLAB CODE BRUTE FORCE m n sizeA n p sizeB C zerosm p for j lzp fori lm for k ln Cia j Cia j AU kB1lt j end end end Page 1 INNER PRODUCT The product for each C 5 is the inner product or dot product of a row vector and a column vector MATLAB CODE m n sizeA n p SizeB C zerosm p for j 1p fori 1m Ciaj AG IBIa j end end Page 2 COLUMN ORIENTED VERSION 0 A b all 6112 39 39 39 aln b1 6 21 6 22 39 39 39 aZn b2 C am1 amZ amn bn all 12 In Cl 21 22 2 c b1 b2 b am1 amZ amn So when 0 is a column vector 0 Akbc Page 3 COLUMN ORIENTED VERSION When C is a 2D matrix C j ZAk Bkj k1 MATLAB CODE m n sizeA n p sizeB C zerosm p forj lzp for k 1n CI j CI j AI1ltquot B1ltj end end or on another level m n sizeA n p sizeB C zerosm p for j lzp CI j Aquot131 j end Page 4 OUTER PRODUCT 1 1V1 1V2 39 39 39 ulvn T 2 2V1 2V2 39 39 39 uZVn v1 v2 n um um v1 um v2 I I I um vn Emphasize row structure of the product ith row llin 1 Y 1V1 1V2 39 39 39 ulvn T T 2 2V1 2V2 39 39 39 u2vn 2 Z T I I I um E umvl umVZ umvn Emphasize column structure of the product jth column 11V 1V1 1V2 39 39 39 ulvn T 2V1 2V2 39 39 39 uZVn W 2M avg an um v1 um v2 I I I um Vn Page 5 OUTER PRODUCT EXAMPLE AB 10 110230 120240 4 310430 320440 30 40 510630 520640 110 120 230 240 310 320 430 440 510 520 630 640 1 2 3 10 20 4 30 40 5 6 A 1B1 A 2B2 B1 3 Alt1 Alt2gtB27 AB is a sum of the outer products formed from the columns of A and the rows of B AB is an inner product of outer products Page 6


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