solving systems of linear equations using substitution worksheet 8 2

Ngày tải lên: 24/10/2014, 21:28

... example, for n = 2 we have p 2s1 p 2s = p 1 2s p 1 2s+1 ,t(p 1 2s )=t(p 2s ),t(p 1 2s+1 )=t(p 2s−1 ),s∈ Z, p 2s p 2s+1 = p 2 2s+1 p 2 2s +2 ,t(p 2 2s+1 )=t(p 2s+1 ),t(p 2 2s +2 )=t(p 2s ),s∈ Z. It ... maps and integrable dynamics, Phys. Lett. A 314 (20 03), 21 4 22 1. (Received October 4, 20 02) ISOMONODROMY TRANSFORMATIONS 1149 Proof. As in the proof of Proposition 1.1, we may assume that n =0. Comparing ... [k 1 ,k 2 ]  ··· [k 2s−1 ,k 2s ]. As a function in the endpoints k 1 , ,k 2s ∈{0, 1, ,N}; this Annals of Mathematics Isomonodromy transformations of linear systems of difference equations ...

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Ngày tải lên: 05/09/2013, 15:28

... 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 = ... 1 -80 0 -87 2- 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 ... RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0- 521 -431 08- 5) Copyright (C) 1 988 -19 92 by Cambridge University Press.Programs Copyright (C) 1 988 -19 92 by Numerical Recipes Software. Permission...

Ngày tải lên: 15/12/2013, 04:15

... 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 ... RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0- 521 -431 08- 5) Copyright (C) 1 988 -19 92 by Cambridge University Press.Programs Copyright (C) 1 988 -19 92 by Numerical Recipes Software. Permission...

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 -431 08- 5) Copyright (C) 1 988 -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 ... ART OF SCIENTIFIC COMPUTING (ISBN 0- 521 -431 08- 5) Copyright (C) 1 988 -19 92 by Cambridge University Press.Programs Copyright (C) 1 988 -19 92 by Numerical Recipes Software. Permission is granted for...

Ngày tải lên: 24/12/2013, 12:16

... 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 ... 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 ... 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

... a N b N      ·      u 1 u 2 ··· u N−1 u N      =      r 1 r 2 ··· r N−1 r N      (2. 4.1) 2. 4 Tridiagonal and Band Diagonal Systems of Equations 51 Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0- 521 -431 08- 5) Copyright (C) 1 988 -19 92 by ... 2. 4 Tridiagonal and Band Diagonal Systems of Equations 53 Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0- 521 -431 08- 5) Copyright (C) 1 988 -19 92 by Cambridge ... ART OF SCIENTIFIC COMPUTING (ISBN 0- 521 -431 08- 5) Copyright (C) 1 988 -19 92 by Cambridge University Press.Programs Copyright (C) 1 988 -19 92 by Numerical Recipes Software. Permission is granted for...

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) ... call 1 -80 0 -87 2- 7 423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America). in terms of which c 11 = Q 1 + Q 4 − Q 5 + Q 7 c 21 = Q 2 + Q 4 c 12 = Q 3 + Q 5 c 22 = Q 1 + ... 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 =...

Ngày tải lên: 24/12/2013, 12:16

... reassuring. 58 Chapter 2. Solution of Linear Algebraic Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0- 521 -431 08- 5) Copyright (C) 1 988 -19 92 by Cambridge ... 56 Chapter 2. Solution of Linear Algebraic Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0- 521 -431 08- 5) Copyright (C) 1 988 -19 92 by Cambridge University ... Machinery). [1] 2. 5 Iterative Improvement of a Solution to Linear Equations 57 Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0- 521 -431 08- 5) Copyright (C) 1 988 -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 -431 08- 5) Copyright (C) 1 988 -19 92 by Cambridge ... 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 -431 08- 5) Copyright (C) 1 988 -19 92 by Cambridge ... RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0- 521 -431 08- 5) Copyright (C) 1 988 -19 92 by Cambridge University Press.Programs Copyright (C) 1 988 -19 92 by Numerical Recipes Software. Permission...

Ngày tải lên: 24/12/2013, 12:16

... 1 2 3 4 5 6 7 8 9 10 11 ija[k] 7 8 8 10 11 12 3 2 4 5 4 sa[k] 3. 4. 5. 0. 5. x 1. 7. 9. 2. 6. (2. 7. 28 ) Here x is an arbitrary value. Notice that, according to the storage rules, the value of ... inverse A −1 → A −1 − z ⊗ w 1+λ (2. 7.5) 2. 7 Sparse Linear Systems 83 Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0- 521 -431 08- 5) Copyright (C) 1 988 -19 92 by Cambridge University ... multiplication of its transpose and a vector. As 2. 7 Sparse Linear Systems 81 Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0- 521 -431 08- 5) Copyright (C) 1 988 -19 92 by Cambridge...

Ngày tải lên: 24/12/2013, 12:16

... attractor NDSolve   4.1540 680 321 294445 1.6 484 28 5 51076 28 2 25 .9 721 0749 686 6006   (13) IDSolve   [ −57.0 724 33 985 95995, 79 . 82 067 089 488 7 58 ] [ 25 3.4101 326 6 12 387 2, 24 2.97953 722 353 984 ] [ 21 0.347760367 329 9, 29 3.06501763 924 2 ... number): Mathematica       1 .81 26 53 486 19 726 65 0.97 089 484 8346366 2. 8. 891065349797593 0.906 326 7430 986 333       (22 ) Algorithm 2       1 .81 26 53 486 4 526 2[63, 76] 0.97 089 484 8 28 5 20 50[0, 6] 1.999999999999999[3, ... are: Mathematica       7 .25 351756 183 085 0 . 82 507349 120 27 126 4.6 486 983 64 72 580 56 −5 .86 54314 28 8 10479 2. 0767 326 60 789 20 9       (24 ) Algorithm 2       7 .25 3517570611[5 42, 623 ] 0 . 82 50734961640 [26 9, 723 ] 4.6 486 983 6 781 35[045, 440] −5 .86 5431440461[656, 85 6] 2. 0767 326 619734[114,...

Ngày tải lên: 12/01/2014, 22:20

... 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 ... +1−j (2 .8. 25 ) Now, starting with the initial values x (1) 1 = y 1 /R 0 G (1) 1 = R −1 /R 0 H (1) 1 = R 1 /R 0 (2 .8. 26 ) we can recurse away. At each stage M we use equations (2 .8. 23 ) and (2 .8. 24 ) ... 92 Chapter 2. Solution of Linear Algebraic Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0- 521 -431 08- 5) Copyright (C) 1 988 -19 92 by Cambridge...

Ngày tải lên: 21/01/2014, 18:20