chapter 2 solution of linear algebraic equations

... equation   a 11 a 12 a 13 a 14 a 21 a 22 a 23 a 24 a 31 a 32 a 33 a 34 a 41 a 42 a 43 a 44   ·     x 11 x 21 x 31 x 41      x 12 x 22 x 32 x 42      x 13 x 23 x 33 x 43      y 11 y 12 y 13 y 14 y 21 y 22 y 23 y 24 y 31 y 32 y 33 y 34 y 41 y 42 y 43 y 44     =     b 11 b 21 b 31 b 41      b 12 b 22 b 32 b 42      b 13 b 23 b 33 b 43      1000 0100 0010 0001     (2. 1.1) Here ... equation   a 11 a 12 a 13 a 14 a 21 a 22 a 23 a 24 a 31 a 32 a 33 a 34 a 41 a 42 a 43 a 44   ·     x 11 x 21 x 31 x 41      x 12 x 22 x 32 x 42      x 13 x 23 x 33 x 43      y 11 y 12 y 13 y 14 y 21 y 22 y 23 y 24 y 31 y 32 y 33 y 34 y 41 y 42 y 43 y 44     =     b 11 b 21 b 31 b 41      b 12 b 22 b 32 b 42      b 13 b 23 b 33 b 43      1000 0100 0010 0001     (2. 1.1) Here ... 36 Chapter 2. Solution of Linear Algebraic Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0- 521 -43108-5) Copyright (C) 1988-19 92 by Cambridge...

Ngày tải lên: 15/12/2013, 04:15

... 1-800-8 72- 7 423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America). Chapter 2. Solution of Linear Algebraic Equations 2. 0 Introduction A set of linear algebraic equations ... this: a 11 x 1 + a 12 x 2 + a 13 x 3 + ···+a 1N x N =b 1 a 21 x 1 + a 22 x 2 + a 23 x 3 + ···+a 2N x N =b 2 a 31 x 1 + a 32 x 2 + a 33 x 3 + ···+a 3N x N =b 3 ··· ··· a M1 x 1 +a M2 x 2 +a M3 x 3 +···+a MN x N = ... improvement of a solution ( 2. 5) ã Various special forms: symmetric positive-denite ( 2. 9), tridiagonal ( 2. 4), band diagonal ( 2. 4), Toeplitz ( 2. 8), Vandermonde ( 2. 8), sparse ( 2. 7) ã Strassens...

Ngày tải lên: 15/12/2013, 04:15

... matrices, a 11 a 12 a 21 a 22 Ã b 11 b 12 b 21 b 22 = c 11 c 12 c 21 c 22 (2. 11.1) Eight, right? Here they are written explicitly: c 11 = a 11 ì b 11 + a 12 ì b 21 c 12 = a 11 ì b 12 + a 12 ì b 22 c 21 = a 21 ì ... set of formulas was, in fact, discovered by Strassen [1] . The formulas are: Q 1 (a 11 + a 22 ) ì (b 11 + b 22 ) Q 2 (a 21 + a 22 ) ì b 11 Q 3 a 11 ì (b 12 b 22 ) Q 4 a 22 ì (b 11 + b 21 ) Q 5 ... b 11 + a 12 ì b 21 c 12 = a 11 ì b 12 + a 12 ì b 22 c 21 = a 21 ì b 11 + a 22 ì b 21 c 22 = a 21 ì b 12 + a 22 ì b 22 (2. 11 .2) Do you think that one can write formulas for the c’s that involve only...

Ngày tải lên: 15/12/2013, 04:15

... 42 Chapter 2. Solution of Linear Algebraic Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0- 521 -43108-5) Copyright (C) 1988-19 92 by Cambridge ... is x i = 1 a  ii   b  i − N  j=i+1 a  ij x j   (2. 2.4) The procedure defined by equation (2. 2.4) is called backsubstitution.Thecom- bination of Gaussian elimination and backsubstitution yields a solution to the set of equations. The ... the case of a single right-hand side vector):    a  11 a  12 a  13 a  14 0 a  22 a  23 a  24 00a  33 a  34 000a  44    ·    x 1 x 2 x 3 x 4    =    b  1 b  2 b  3 b  4    (2. 2.1) Here...

Ngày tải lên: 24/12/2013, 12:16

... β 22 β 23 β 24 00β 33 β 34 000β 44   =   a 11 a 12 a 13 a 14 a 21 a 22 a 23 a 24 a 31 a 32 a 33 a 34 a 41 a 42 a 43 a 44   (2. 3 .2) We can use a decomposition such as (2. 3.1) to solve the linear set A · x =(L·U)·x=L·(U·x)=b (2. 3.3) by ... j : α i1 β 1j + α i2 β 2j + ···+α ii β jj = a ij (2. 3.9) i>j: α i1 β 1j +α i2 β 2j +···+α ij β jj = a ij (2. 3.10) Equations (2. 3.8)– (2. 3.10) total N 2 equations for the N 2 + N unknown α’s ... case of a 4 ì 4 matrix A, for example, equation (2. 3.1) would look like this:   α 11 000 α 21 α 22 00 α 31 α 32 α 33 0 α 41 α 42 α 43 α 44   ·   β 11 β 12 β 13 β 14 0 β 22 β 23 β 24 00β 33 β 34 000β 44   =   a 11 a 12 a 13 a 14 a 21 a 22 a 23 a 24 a 31 a 32 a 33 a 34 a 41 a 42 a 43 a 44   (2. 3 .2) We...

Ngày tải lên: 24/12/2013, 12:16

... elimination. dum=a[i][1]/a[k][1]; al[k][i-k]=dum; for (j =2; j<=mm;j++) a[i][j-1]=a[i][j]-dum*a[k][j]; a[i][mm]=0.0; } } } 52 Chapter 2. Solution of Linear Algebraic Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC ... 1967, Computer Solution of Linear Algebraic Systems (Engle- wood Cliffs, NJ: Prentice-Hall), Chapters 9, 16, and 18. Westlake, J.R. 1968, A Handbook of Numerical Matrix Inversion and Solution of Linear Equations (New ... Computations , 2nd ed. (Baltimore: Johns Hopkins University Press), Đ 4.3. 50 Chapter 2. Solution of Linear Algebraic Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC...

Ngày tải lên: 24/12/2013, 12:16

... Inverse(a 11 ) R 2 = a 21 ì R 1 R 3 = R 1 ì a 12 R 4 = a 21 ì R 3 R 5 = R 4 a 22 R 6 = Inverse(R 5 ) c 12 = R 3 ì R 6 c 21 = R 6 ì R 2 R 7 = R 3 ì c 21 c 11 = R 1 R 7 c 22 = R 6 (2. 11.6) ... matrices  a 11 a 12 a 21 a 22  and  c 11 c 12 c 21 c 22  (2. 11.5) are inverses of each other. Then the c’s can be obtained from the a’s by the following operations (compare equations 2. 7 .22 and 2. 7 .25 ): R 1 = Inverse(a 11 ) R 2 = a 21 ì R 1 R 3 = ... 104 Chapter 2. Solution of Linear Algebraic Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0- 521 -43108-5) Copyright (C) 1988-19 92 by Cambridge...

Ngày tải lên: 24/12/2013, 12:16

... 56 Chapter 2. Solution of Linear Algebraic Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0- 521 -43108-5) Copyright (C) 1988-19 92 by Cambridge ... quite far from the true solution, one call is generally enough; but a second call to verify convergence can be reassuring. 58 Chapter 2. Solution of Linear Algebraic Equations Sample page from ... feasible)? 2. 5 Iterative Improvement of a Solution to Linear Equations 55 Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0- 521 -43108-5) Copyright (C) 1988-19 92 by Cambridge...

Ngày tải lên: 24/12/2013, 12:16

... (scale) { 62 Chapter 2. Solution of Linear Algebraic Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0- 521 -43108-5) Copyright (C) 1988-19 92 by Cambridge ... q.e.d. Figure 2. 6.1 summarizes our discussion of SVD thus far. 68 Chapter 2. Solution of Linear Algebraic Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0- 521 -43108-5) Copyright ... n ≤ N (2. 6.3) 66 Chapter 2. Solution of Linear Algebraic Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0- 521 -43108-5) Copyright (C) 1988-19 92 by Cambridge...

Ngày tải lên: 24/12/2013, 12:16

... 0. 0. 0. 2. 0. 0. 0. 6. 5.      (2. 7 .27 ) 74 Chapter 2. Solution of Linear Algebraic Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0- 521 -43108-5) Copyright ... 1978 (Philadelphia: S.I.A.M.). 82 Chapter 2. Solution of Linear Algebraic Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0- 521 -43108-5) Copyright (C) 1988-19 92 by Cambridge ... j,jl,jm,jp,ju,k,m,n2,noff,inc,iv; float v; n2=ija[1]; Linear size of matrix plus 2. for (j=1;j<=n2 -2; j++) sb[j]=sa[j]; Diagonal elements. iindexx(ija[n2-1]-ija[1],(long *)&ija[n2-1],&ijb[n2-1]); Index...

Ngày tải lên: 24/12/2013, 12:16

... 92 Chapter 2. Solution of Linear Algebraic Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0- 521 -43108-5) Copyright (C) 1988-19 92 by Cambridge ... equations      11··· 1 x 1 x 2 ··· x N x 2 1 x 2 2 ··· x 2 N ··· x N−1 1 x N−1 2 ··· x N−1 N      ·      w 1 w 2 w 3 ··· w N      =      q 1 q 2 q 3 ··· q N      (2. 8 .2) Write this ... x j .Inotherwords, P j (x i )=δ ij = N  k=1 A jk x k−1 i (2. 8.4) But (2. 8.4) saysthat A jk is exactly the inverse of the matrix of componentsx k−1 i ,which appears in (2. 8 .2) , with the subscript as the column index. Therefore the solution of (2. 8 .2) is...

Ngày tải lên: 21/01/2014, 18:20

... +1,i +2, ,N (2. 9.5) 92 Chapter 2. Solution of Linear Algebraic Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0- 521 -43108-5) Copyright (C) 1988-19 92 by ... by L T ij = L ji (2. 9.3) Writing out equation (2. 9 .2) in components, one readily obtains the analogs of equations (2. 3. 12) – (2. 3.13), L ii =  a ii − i−1  k=1 L 2 ik  1 /2 (2. 9.4) and L ji = 1 L ii  a ij − i−1  k=1 L ik L jk  j ... 96 Chapter 2. Solution of Linear Algebraic Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0- 521 -43108-5) Copyright (C) 1988-19 92 by Cambridge...

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... of the outreach of bank- ing outlets. 25 Chapter 2: Indicators of Financial Structure, Development, and Soundness 1 I H G F E D C B A 12 11 10 9 8 7 6 5 4 3 2 distribution of stress tests results, ... other indicators of competition. See chapter 4 for a discussion of model-based indicators of competition. 6. For an example of the computation of the Herfindahl index, see chapter 15 of the Compilation ... discussion of these indicators can be found in section 2. 2.4. Table 2. 2 summarizes the indicators of financial system performance that have been discussed in this section. 2. 1.4 Scope and Coverage of...

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... Đ1.8 Chapter 2 Chapter 3 Chapter 5 Đ6.1 Đ6 .2 Đ6.3 Đ7.1 Đ7 .2 Đ7.3 Đ7.4 Đ8.1 Đ8 .2 Đ8.3 Đ8.4 Đ8.5 Đ8.6 Đ8.8 Chapter 9 Chapter 10 Chapter 11 Đ 12. 1 Đ 12. 2 Đ13.1 Đ13 .2 Đ13.3 Đ13.4 Đ13.7 Chapter ... proofs the selected material provides the underlying mathe- matics. Chapter 1 Chapter 2 Chapter 4 Chapter 5 Đ6.1 Đ6 .2 Đ6.3 Đ7.1 Đ7 .2 Đ7.3 Đ7.4 Đ8.1 Đ8 .2 Đ8.3 Đ8.4 Đ8.5 Đ8.6 Đ8.8 Đ8.9 Chapter ... background: Đ1.1 Đ1 .2 Đ1.3 Đ1.4 Đ1.5 Đ1.6 Đ1.7 Đ1.8 Đ5.1 Đ5 .2 Đ5.3 Đ5.4 Đ6.1 Đ6 .2 Đ6.3 Đ7.1 Đ7 .2 Đ7.3 Đ7.4 Đ8.1 Đ8 .2 Đ8.3 Đ8.4 Đ8.5 Đ8.6 Đ8.8 Đ11.1 Đ11 .2 Đ11.3 Đ11.4 Đ11.5 Đ 12. 1 Đ 12. 2 Đ13.1 Đ13 .2 Đ13.3...

Ngày tải lên: 19/02/2014, 22:20

... Copyright â 20 06 Pearson Addison- Wesley. All rights reserved. 2- 9 Display 2. 3 Precedence of Operators (4 of 4) Copyright â 20 06 Pearson Addison- Wesley. All rights reserved. 2- 5 Evaluating ... Display 2. 2 Truth Tables Copyright â 20 06 Pearson Addison- Wesley. All rights reserved. 2- 33 Comma Operator Evaluate list of expressions, returning value of the last expression ♦ Most often ... Addison- Wesley. All rights reserved. 2- 8 Display 2. 3 Precedence of Operators (3 of 4) Copyright â 20 06 Pearson Addison- Wesley. All rights reserved. 2- 12 if-else Statement Syntax ♦ Formal syntax: if...

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