J. W. Daniel, "The conjugate gradient method for linear and nonlinear operator equations," SIAM J. Numer. Anal., vol. 4, pp. 10--26, 1967.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....m,i s m (# i ) # 1 2d# ) Now c k (s) cosh(k cosh 1 s) We use the formulas cosh x = e x e x (8.3) and ln z = cosh 1 when x =lnz (8.4) to determine our upper bound as outlined below. We note that this type of proof technique was used at least as early as the 1960s (see [3]) 1) Find a z such that ( z z 1 2 ) s m,i ,i m. 2) Use z and (8.3) to determine c k (s m,i ) i m. 3) Find an upper bound for c k (s m,i ) i m. 4) Use this bound and (8.2) to deduce an upper bound for p k,m (s m,i ) i m. 5) Use the above information to obtain the upper bound on ....

J. W. Daniel, The conjugate gradient method for linear and nonlinear operator equations, SIAM J. Numer. Anal., 4 (1967), pp. 10--26.

....k = 1; g k k d k 1 ; for k 2, 0.3) where g k = rf(x k ) k is a stepsize obtained by a one dimensional line search and k is a scalar. Since Fletcher and Reeves introduced the nonlinear conjugate gradient method in 1964, many formulae have been proposed to compute the scalar k , see [1, 2, 3, 4, 5, 6, 7, 8, 9] etc. Among them, two well known formulae for k are called the FR and PRP formulae (see [4, 7, 8] and are given by =kg k 1 k (0.4) PRP k y k 1 =kg k 1 k (0.5) respectively, where y k 1 = g k g k 1 and k k means the Euclidean norm. The properties of nonlinear conjugate gradient ....

Daniel, J. W., The conjugate gradient method for linear and nonlinear operator equations, SIAM J. Numer. Anal., 1967, 4: 10-26.

....If the eigenvalues fffi j g of B B are such that 0 ffi 1 : ffi p Gamma1 ffi p j b 1 : ffi n2 Gammaq j b 2 ffi n2 Gammaq 1 : ffi n2 ; n2 j=n2 Gammaq 1 9 = 3.8. 1) b j 1: 2 The proof can be found in Van der Vorst [18] see also Daniel [6]. Notice that for n 2 Gamma q 1 j n 2 and ffi 2 [b 1 ; b 2 ] we have, 0 1. Thus (3.8.1) can be rewritten as : 3.8.2) From lemmas 3.5.5, we see that if ff 1 or if ff = 1 such that (3.5.6) holds, then ffi j C Delta h 2 , for 1 j n 2 . Thus for 1 j p and ffi 2 [b ....

Daniel, J., The Conjugate Gradient Method for Linear and Nonlinear Operator Equations. SIAM J. Numer. Anal., V4, 1967, pp. 10-26.

....i can be expressed as a polynomial in A of degree i Gamma 1, acting on r 0 . The minimization interpretation makes it possible to bound the error for CG by replacing the CG polynomial by easier to analyze polynomials, for instance a Chebyshev polynomial. This leads to the well known upper bound [102, 48, 45, 4] kx i Gamma xkA p 2 p Gamma 1 p 1 i kx 0 Gamma xkA ; 5) for symmetric positive definite matrices, in which = max (A) min (A) This upper bound describes well the convergence behavior for matrices A of which the eigenvalues are distributed rather homogeneously. For more ....

J. W. Daniel. The conjugate gradient method for linear and nonlinear operator equations. SIAM J. Numer. Anal., 4:10--26, 1967. 21

....of formulae (1.8) and (2.1) in computations is observed in [22] that is, the next direction d k 1 in (1.2) is independent of the length of d k when fi k takes the form of (1.8) or (2. 1) Many authors have presented other choices for the scalar fi k , for example Buckley and Lenir [2] Daniel [13], Gilbert and Nocedal [16] Qi et al. [28] Shanno [29] and Touati Ahmed and Storey [30] Observing that the formulae (1.6) 1.9) 1.12) and (2.1) share two nominators and three denominators, we can use the combinations of these nominators and denominators to obtain the following ....

J. W. Daniel, The conjugate gradient method for linear and nonlinear operator equations, SIAM J. Numer. Anal., 4 (1967), 10-26.

....can be expressed as a polynomial in A of degree i Gamma 1, acting on r 0 . The minimization interpretation makes it possible to bound the error for CG by replacing the CG polynomial , by easier to analyse polynomials, for instance a Chebyshev polynomial. This leads to the well known upper bound [99, 43, 40, 4] kx i Gamma xkA p 2 p Gamma 1 p 1 i kx i Gamma xkA ; 5) for symmetric positive definite matrices, in which = max (A) min (A) This upper bound describes well the convergence behavior for matrices A of which the eigenvalues are distributed rather homogeneously. For more uneven ....

J. W. Daniel. The conjugate gradient method for linear and nonlinear operator equations. SIAM J. Numer. Anal., 4:10--26, 1967.

....iterative methods which coincide with known iterative methods for linear systems. Some nonlinear iterative methods have been derived, studied and used in various applications for steepest descent methods, SOR type and conjugate gradient type methods for nonsymmetric Jacobians (see [4] 5] 6] [7], 8] 12] 14] 15] 16] 17] The Generalized Conjugate Gradients (GCG) see [1] is an iterative method applicable to nonsymmetric linear systems. The main goal of this article is to derive nonlinear versions of GCG and to establish global convergence results. In Sect. 2, we derive the ....

Daniel, J.W. (1967): The conjugate gradient method for linear and nonlinear operator equations. SIAM J. Numer. Anal. 4, 10--26

....continuous least squares Hessian, perhaps the most obvious approach is to precondition with L, converting (3.1) to standard form [10] H = H ls ffI ; H ls a discretization of a compact operator: 3. 2) In this case, the rate of convergence of PCG is known to be asymptotically superlinear [5] as the discretization level h 0. Since L has a nontrivial null space, a preliminary transformation is required. Consider the decomposition s = av 0 s ; 3.3) where s 2 null(L) and v 0 2 null(L) with jjv 0 jj = 1. From the linearizations (2.3) and (2.14) one obtains the quadratic ....

....Hessian (bottom row of subplots) for ff = 1 (left subplots) and ff = 10 Gamma3 (right subplots) cost, since a several systems (3.1) with relatively large ff must be solved. We close this section with an explanation for the deterioration in PCG performance as ff decreases. The analysis in [5] shows that CG is superlinearly convergent for Hilbert space operators equations in which the operator is a compact perturbation of the identity. Unfortunately, this analysis is qualitative and gives no indication of the effects of varying parameters like ff. Looking at the continuous version of ....

J. W. Daniel, The conjugate gradient method for linear and nonlinear operator equations, SIAM J. Numer. Anal., Vol. 4 (1967), pp. 10-26.

....but it had not yet been recognized as a relative of the conjugate gradient algorithm that generates the same sequence of iterates when applied to quadratic minimization. Major steps in understanding the convergence behavior of the conjugate gradient algorithm were made by Kaniel [34] and Daniel [12]. Although both papers contained some errors [4, 13, 8] the standard bounds on the convergence rate for conjugate gradients, derived using Chebyshev polynomials, can be found here. Thus the 1960 s brought considerable progress in the use and understanding of the KMP family. 4 The 1970 s The key ....

J. W. Daniel, The conjugate gradient method for linear and nonlinear operator equations, SIAM J. Numer. Anal., 4 (1967a), pp. 10--26.

....requirements are those of T , K and four vectors. The preconditioned CG algorithm develops an approximate solution of the form z k 1 = z 0 k Gamma K Gamma1 T Delta K Gamma1 T (z Gamma z 0 ) k = 0; 1; 2; 3:31) where k (x) is a polynomial of degree k. It is known ([23], x1.2; 67] x4) that if T is a Stieltjes matrix and K is symmetric and positive definite, the preconditioned CG algorithm minimizes kz Gamma z k k T over all algorithms of the form (3.31) Prior to each invocation of the CG solver, BLIP checks that T is positive definite using Theorem 3.5 as a ....

J.W. Daniel. The conjugate gradient method for linear and nonlinear operator equations. SIAM Journal of Numerical Analysis, 4:10--26, 1967.

....k (s) cosh(k cosh Gamma1 s) We use the formulas cosh x = e x e Gammax 2 (9.3) and ln z = cosh Gamma1 z z Gamma1 2 when x = ln z (9.4) to determine our upper bound as outlined below. We note that this type of proof technique was used at least as early as the 1960 s (see [3]) 1. Find a z such that i z z Gamma1 2 j = s m;i ; i m. 2. Use z and (9.3) to determine c k (s m;i ) i m. 3. Find an upper bound for c 2 k (s m;i ) i m. 4. Use this bound and (9.2) to deduce an upper bound for p 2 k;m (s m;i ) i m. 5. Use the above information to obtain ....

J. W. Daniel. The conjugate gradient method for linear and nonlinear operator equations. SIAM Journal on Numerical Analysis, 4:10--26, 1967.

....using a Gauss transform. In the final section, we collect the error bounds known to us. The analysis is based on best polynomial approximation problems on the spectrum of the operator which defines the Krylov subspaces. Such a result is well known for the positive definite case; cf. Daniel [5] and Luenberger [30] The other cases are now also well understood primarily because of the efforts of Freund [14,15] and Freund and Ruscheweyh [16] We also note that there are other conjugate gradient type methods for general nonsymmetric problems that can be characterized variationally, but for ....

...., the space of polynomials of degree k Gamma 1. The errors of the conjugate gradient methods considered in this paper can be estimated from above in terms of the solution of certain best polynomial approximation problems. This is well known for the standard conjugate gradient method; cf. Daniel [5] and Luenberger [30] As we will see, the same estimate can also be used for several other cases, but for others, more complicated polynomial approximation problems arise. These problems are also quite well understood, primarily because of the efforts of Freund [15] and Freund and Ruscheweyh [16] ....

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James W. Daniel. The conjugate gradient method for linear and nonlinear operator equations. SIAM Journal on Numerical Analysis, 4:10--26, 1967.

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J. W. Daniel, "The conjugate gradient method for linear and nonlinear operator equations," SIAM J. Numer. Anal., vol. 4, pp. 10--26, 1967.

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J. W. DANIEL, The conjugate gradient method for linear and nonlinear operator equations, SIAM J. Numer. Anal., 4 (1967), pp. 10--26.

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J. W. Daniel, The conjugate gradient method for linear and nonlinear operator equations, SIAM J. Numer. Anal., 4 (1967), pp. 10--26.

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Daniel, J.W., The conjugate gradient method for linear and nonlinear operator equations, SIAM J. Numer. Anal., 4 (1967) No.1., 10-26.

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J. W. Daniel, The conjugate gradient method for linear and nonlinear operator equations, SIAM J. Numer. Anal., 4(1) (1967), pp. 10--26.

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J. Daniel, The Conjugate Gradient Method for Linear and Nonlinear Operator Equations, SIAM J. Numer. Anal., 4 (1967), pp. 10--26.

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