3 solution of nonlinear simultaneous equations multidimensional newton raphson method

... For the case of a × matrix A, for example, equation (2 .3. 1) would look like this:       α11  α21 31 α41 α22 32 α42 0 33 α 43 β11   · 0 α44 β12 β22 0 β 13 β 23 33 β14 β24  34 β44 a11 ... =  21 a31 a41 a12 a22 a32 a42 a 13 a 23 a 33 a 43 a14 a24  a34 a44 (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 first ... Chapter Solution of Linear Algebraic Equations Backsubstitution But how we solve for the x’s? The last x (x4 in this example) is already isolated, namely x4 = b4 /a44 (2.2.2) x3 = [b − x4 a34 ] a33...

Ngày tải lên: 24/12/2013, 12:16

... impulsive differential equations, ” Nonlinear Analysis: Theory, Methods & Applications, vol 25, no 4, pp 32 7 33 7, 1995 D Guo, “Existence of solutions of boundary value problems for nonlinear second ... “Multiple positive solutions for first order nonlinear impulsive integro-differential equations in a Banach space,” Applied Mathematics and Computation, vol 1 43, no 2 -3, pp 233 –249, 20 03 33 J Li and J ... impulsive nonlinear Fredholm integral equations and applications,” Journal of Mathematical Analysis and Applications, vol 1 73, no 1, pp 31 8 32 4, 19 93 D Guo and X Liu, “Multiple positive solutions of...

Ngày tải lên: 21/06/2014, 11:20

... ∈ RN The proof of Lemma 3. 1 is given for completeness in the appendix of this article Lemma 3. 2 is an extension of [17, Lemma 1], so the proof is omitted here for briefness Proof of Theorem 2.1 ... Analysis, Theory, Methods & Applications 33 (1998), no 3, 211– 234 [5] P de Mottoni, A Schiaffino, and A Tesei, Attractivity properties of nonnegative solutions for a class of nonlinear degenerate ... Applicata, Serie Quarta 136 (1984), 35 –48 [6] A L Edelson, Entire solutions of singular elliptic equations, Journal of Mathematical Analysis and Applications 139 (1989), no 2, 5 23 532 [7] N Fukagai,...

Ngày tải lên: 22/06/2014, 22:20

... corresponding x’s are then the columns of the matrix inverse of A (§2.1 and §2 .3) • Calculation of the determinant of a square matrix A (§2 .3) If M < N , or if M = N but the equations are degenerate, then ... occur during the solution In fact, the preceding item can be viewed as the special case where the loss of significance is unfortunately total 34 Chapter Solution of Linear Algebraic Equations at ... effectively fewer equations than unknowns In this case there can be either no solution, or else more than one solution vector x In the latter event, the solution space consists of a particular solution...

Ngày tải lên: 15/12/2013, 04:15

... b21  b31 b41   b12  b22  b32 b42    x12  x22  x32 x42  x 13  x 23  x 33 x 43  b 13  b 23  b 33 b 43  0 0 y11  y21 y31 y41 0 0 y12 y22 y32 y42 y 13 y 23 y 33 y 43  y14 y24  y34 y44 ... 37 2.1 Gauss-Jordan Elimination Elimination on Column-Augmented Matrices  Consider the linear matrix equation      a12 a22 a32 a42 a 13 a 23 a 33 a 43 a14 x11 a24   x21  · a34 x31 a44 ... looks like this (in the case of a single right-hand side vector):       x1 b1 a11 a12 a 13 a14  a22 a 23 a24   x2   b2  (2.2.1) ·  =    0 a 33 a34 x3 b3 0 a44 x4 b4 Here the primes...

Ngày tải lên: 15/12/2013, 04:15

... part (2.10.4) of the algorithm is needed, so we separate it off into its own routine rsolv Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521- 431 08-5) Copyright ... Instead of doing O(N ) operations each time to solve the equations from scratch, one can often update a matrix factorization in O(N ) operations and use the new factorization to solve the next set of ... float **qt, int n, int i, float a, float b); int i,j,k; 102 Chapter Solution of Linear Algebraic Equations We will make use of QR decomposition, and its updating, in §9.7 CITED REFERENCES AND...

Ngày tải lên: 15/12/2013, 04:15

... (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  α21  31 α41 β12 β22 32 α42 β 13 β 23 33 α 43  β14 β24   34 β44 (2 .3. 14) by ... αij βjj = aij (2 .3. 8) (2 .3. 9) (2 .3. 10) Equations (2 .3. 8)–(2 .3. 10) total N equations for the N + N unknown α’s and β’s (the diagonal being represented twice) Since the number of unknowns is greater ... is i = j of equation (2 .3. 12) and i = j +1 N sum=a[i][j]; of equation (2 .3. 13) for (k=1;k

Ngày tải lên: 24/12/2013, 12:16

... improved solution x 2.5 Iterative Improvement of a Solution to Linear Equations Obviously it is not easy to obtain greater precision for the solution of a linear set than the precision of your ... Unfortunately, for large sets of linear equations, it is not always easy to obtain precision equal to, or even comparable to, the computer’s limit In direct methods of solution, roundoff errors accumulate, ... number of right-hand sides can be solved in turn by repeated calls to banbks, the backsubstitution routine whose analog in §2 .3 is lubksb 55 2.5 Iterative Improvement of a Solution to Linear Equations...

Ngày tải lên: 24/12/2013, 12:16

... or call 1-800-872-74 23 (North America only),or send email to trade@cup.cam.ac.uk (outside North America) c22 = Q1 + Q3 − Q2 + Q6 104 Chapter Solution of Linear Algebraic Equations CITED REFERENCES ... Mathematik, vol 13, pp 35 4 35 6 [1] Kronsjo, L 1987, Algorithms: Their Complexity and Efficiency, 2nd ed (New York: Wiley) ¨ Winograd, S 1971, Linear Algebra and Its Applications, vol 4, pp 38 1 38 8 Pan, ... each hierarchical level of the recursion In total it reduces the process of matrix multiplication to order N log2 instead of N What about all the extra additions in (2.11 .3) –(2.11.4)? Don’t they...

Ngày tải lên: 24/12/2013, 12:16

... names, including Schultz’s Method and Hotelling’s Method; see Pan and Reif [1] for references In fact, equation (2.5.11) is simply the iterative Newton- Raphson method of root-finding (§9.4) applied ... inverse of the matrix A, so that B0 · A is approximately the identity matrix Define the residual matrix R of B0 as 58 Chapter Solution of Linear Algebraic Equations We can define the norm of a matrix ... discussion of the use of SVD in this application to Chapter 15, whose subject is the parametric modeling of data SVD methods are based on the following theorem of linear algebra, whose proof is beyond...

Ngày tải lên: 24/12/2013, 12:16

... solutions of A⋅x = d solutions of A ⋅ x = c′ SVD solution of A ⋅ x = c range of A d c′ c SVD solution of A⋅x = d (b) Figure 2.6.1 (a) A nonsingular matrix A maps a vector space into one of the ... same permutation of the columns of U, elements of W, and columns of V (or rows of VT ), or (ii) forming linear combinations of any columns of U and V whose corresponding elements of W happen to ... combination of the set of equations that we are trying to solve The resolution of the paradox is that we are throwing away precisely a combination of equations that is so corrupted by roundoff error...

Ngày tải lên: 24/12/2013, 12:16

... applications.) • Each of the first N locations of ija stores the index of the array sa that contains the first off-diagonal element of the corresponding row of the matrix (If there are no off-diagonal elements ... parameter γ = −b1 to avoid loss of precision by subtraction in the first of equations (2.7.11) In the unlikely event that this causes loss of precision in the second of these equations, you can make a ... to successive minimizations of the function 86 Chapter Solution of Linear Algebraic Equations and modifies the definitions of αk , βk , pk , and pk in equation (2.7 .32 ): rk · zk pk · A · pk rk+1...

Ngày tải lên: 24/12/2013, 12:16

... Handbook of Numerical Matrix Inversion and Solution of Linear Equations (New York: Wiley) [2] von Mises, R 1964, Mathematical Theory of Probability and Statistics (New York: Academic Press), pp 39 4ff ... related to those of L by LT = Lji ij (2.9 .3) Writing out equation (2.9.2) in components, one readily obtains the analogs of equations (2 .3. 12)–(2 .3. 13) , 1/2 i−1 Lii = aii − L2 ik (2.9.4) k=1 ... the given values qj of the first N moments (For more on this problem, consult [3] .) The routine given in this section solves (2.8.2) 92 Chapter Solution of Linear Algebraic Equations The routine...

Ngày tải lên: 21/01/2014, 18:20

... Handbook of Numerical Matrix Inversion and Solution of Linear Equations (New York: Wiley) [2] von Mises, R 1964, Mathematical Theory of Probability and Statistics (New York: Academic Press), pp 39 4ff ... related to those of L by LT = Lji ij (2.9 .3) Writing out equation (2.9.2) in components, one readily obtains the analogs of equations (2 .3. 12)–(2 .3. 13) , 1/2 i−1 Lii = aii − L2 ik (2.9.4) k=1 ... systems of linear equations However, we will meet special cases where QR is the method of choice Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521- 431 08-5)...

Ngày tải lên: 21/01/2014, 18:20

... 8.9 Approximation of Pure Jump Processes 8.10 Exercises 34 7 34 7 35 0 35 5 35 6 35 9 36 1 36 2 36 8 37 5 38 8 Estimating Discretely ... Solutions of SDEs 1.10 Exercises 1 16 23 26 34 38 45 53 57 59 Exact Simulation of Solutions of SDEs 2.1 Motivation of ... → §6.2 → §6 .3 ↓ §7.1 → §7.2 → §7 .3 → §7.4 ↓ §8.1 → §8.2 → §8 .3 → §8.4 → §8.5 → §8.6 → §8.8 ↓ Chapter ↓ Chapter 10 ↓ Chapter 11 ↓ §12.1 → §12.2 ↓ § 13. 1 → § 13. 2 → § 13. 3 → § 13. 4 → § 13. 7 ↓ Chapter...

Ngày tải lên: 19/02/2014, 22:20

... 3. 3507 4.6806 4. 839 1 4 .34 03 ⎟ ⎜ ⎟ B3 = ⎜ 3. 3102 4. 839 1 5.2014 4. 739 6 ⎟ ⎜ ⎟ ⎝ 2.9201 4 .34 03 4. 739 6 4 .37 50 ⎠ 0 0 We use the iterative algorithm (14) to solve (12) for different values of X1(0): ⎞ ⎛ ... (X3 + B3 )1/4 A3 , ⎪ ⎨ 1/5 1 /3 1/4 X = In + A∗ (X1 + B1 )1/4 A1 + A∗ (X2 + B2 )1/2 A2 + A∗ (X3 + B3 )1 /3 A3 , ⎪ ⎪ ⎩ 1/4 1/5 1 /3 X3 = In + A∗ (X1 + B1 )1 /3 A1 + A∗ (X2 + B2 )1/4 A2 + A∗ (X3 + B3 ... 2.7 937 X1 (15) = ⎝ −4.4081 16.8 83 −6.6118 ⎠ , 2.7 937 −6.6118 9.7152 ⎛ ⎞ 11.512 −5.8429 3. 1922 X2 (15) = ⎝ −5.8429 19.485 −7. 930 8 ⎠ 3. 1922 −7. 930 8 10.68 and ⎞ 11. 235 3. 5241 3. 2712 X3 (15) = ⎝ 3. 5241...

Ngày tải lên: 20/06/2014, 22:20

... n2 = max0 ≤ t,s ≤1 G2(t, s) Then the conclusion of Theorem 3. 3 also holds Proof This proof is similar to that of Theorem 3. 3, so is omitted Remark 3. 2 In this article, conditions f(t, 0) ≡ and ... existence and uniqueness of solutions for initial value problem of fractional differential equations J Univ Jinan 2010, 24 :31 2 -31 5 Page 12 of 13 Zhao et al Advances in Difference Equations 2011, 2011:10 ... of Arzela-Ascoli Theorem, T’: P’ ® P’ is completely continuous Remark 3. 1 Theorems 3. 1 and 3. 2 also hold for the boundary value problem (1.1) Proof This proof is similar to that of Theorems 3. 1...

Ngày tải lên: 21/06/2014, 03:20

... k→∞ u0 in E It is easy to ∀n ∈ Z 3. 32 Hence, we have, by 3. 31 and 3. 32 , u0 ∞ ≤ A 3. 33 By F5 , there exists M2 > such that ϕ |x| ≤ M2 |x|γ1 −1 , ∀x ∈ R, |x| ≤ A 3. 34 For any given number ε > 0, ... |uk n − u0 n | < ε, for k ≥ k0 3. 36 16 Advances in Difference Equations On the other hand, it follows from 3. 31 , 3. 33 , 3. 34 , 3. 35 , and F5 that − f n, u0 n f n, uk n |uk n − u0 n | |n|>Π ≤ ... solution The proof is complete Proof of Theorem 1.2 In the proof of Theorem 1.1, the condition that F2 n, x ≥ for n, x ∈ J × R, |x| ≤ in F1 , is only used in the the proofs of assumption ii of...

Ngày tải lên: 21/06/2014, 11:20

... positive solution Proof The proof is similar to that of Theorem 3. 1 and is therefore omitted Theorem 3. 3 If H1 f t, x, y < x y for t ∈ 0, and x y ∈ 0, ∞ , then the BVP 1 .3 has no monotone positive solution ... ε1 ≤ 3. 2 By the definition of f , we may choose ρ1 > so that f t, x, y ≤ f Let Ω1 ε1 x y , for t ∈ 0, , x y ∈ 0, ρ1 3. 3 {u ∈ E : u < ρ1 /2} Then for any u ∈ K ∩ ∂Ω1 , in view of 3. 2 and 3. 3 , ... from 3. 6 , 3. 10 , and Theorem 1.1 that the operator T has one fixed point u ∈ K ∩ Ω2 \ Ω1 , which is a monotone positive solution of the BVP 1 .3 Theorem 3. 2 If H1 f ∞ < < H2 f0 , then the BVP 1.3...

Ngày tải lên: 21/06/2014, 11:20

... r 3. 40 r↓0 Theorem 3. 10 Perron’s method Let comparison hold for 3. 2 ; that is, if w is a subsolution of 3. 2 and v is a supersolution of 3. 2 , then w ≤ v Suppose also that there is a subsolution ... Problems 11 a supersolution u of 3. 2 that satisfies the boundary condition u∗ x W x u∗ x for x ∈ ∂Ω Then sup w x : u ≤ w ≤ u and w is a subsolution of 3. 2 3. 41 is a solution of 3. 2 From paper , ... Problems the proof of existence of viscosity solutions of first-order Hamilton-Jacobi equations, Crandall et al had applications of Perron’s Method to second-order partial differential equations in...

Ngày tải lên: 21/06/2014, 20:20