2.3 LU Decomposition and Its Applications 45 a etc. http://www.nr.com or call 1-800-872- (including this one) internet ( ① diagonal elements subdiagonal elements -7423(North America to any server computer,is users to make one paper etc. only),or send to dir Copyright (C) from NUMERICAL RECIPES IN C:THE ART OF SCIENTIFIC COMPUTING(ISBN 1988-1992 by Cambridge University Press.Programs Figure 2.3.1.Crout's algorithm for LU decomposition of a matrix.Elements of the original matrix are modified in the order indicated by lower case letters:a,b,c.etc.Shaded boxes show the previously modified elements that are used in modifying two typical elements,each indicated by an"x". 1788-1982 If you work through a few iterations of the above procedure,you will see that the a's and B's that occur on the right-hand side of equations(2.3.12)and(2.3.13) .Further reproduction, Numerical Recipes 10-621 are already determined by the time they are needed.You will also see that every a is used only once and never again.This means that the corresponding a ij or Bij can be stored in the location that the a used to occupy:the decomposition is"in place." 43195 [The diagonal unity elements a(equation 2.3.11)are not stored at all.]In brief, (outside Crout's method fills in the combined matrix of a's and B's. North Software. 11 012613 3141 ing of 021 322323 524 (2.3.14) Q31 032 333 34 visit website machine L041 Q42 043 344」 by columns from left to right,and within each column from top to bottom (see Figure 2.3.1). What about pivoting?Pivoting(i.e.,selection of a salubrious pivot element for the division in equation 2.3.13)is absolutely essential for the stability of Crout's2.3 LU Decomposition and Its Applications 45 Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machine￾readable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). c g i b d f h j diagonal elements subdiagonal elements etc. etc. x x a e Figure 2.3.1. Crout’s algorithm for LU decomposition of a matrix. Elements of the original matrix are modified in the order indicated by lower case letters: a, b, c, etc. Shaded boxes show the previously modified elements that are used in modifying two typical elements, each indicated by an “x”. If you work through a few iterations of the above procedure, you will see that the α’s and β’s that occur on the right-hand side of equations (2.3.12) and (2.3.13) are already determined by the time they are needed. You will also see that every a ij is used only once and never again. This means that the corresponding α ij or βij can be stored in the location that the a used to occupy: the decomposition is “in place.” [The diagonal unity elements αii (equation 2.3.11) are not stored at all.] In brief, Crout’s method fills in the combined matrix of α’s and β’s,    β11 β12 β13 β14 α21 β22 β23 β24 α31 α32 β33 β34 α41 α42 α43 β44    (2.3.14) by columns from left to right, and within each column from top to bottom (see Figure 2.3.1). What about pivoting? Pivoting (i.e., selection of a salubrious pivot element for the division in equation 2.3.13) is absolutely essential for the stability of Crout’s