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## Stair Matrices And Their Generalizations With Applications To Iterative Methods I: A Generalization Of The Sor Method (0)

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654 |
Iterative Solution Methods
- Axelsson
- 1993
(Show Context)
Citation Context ... r = #/2 and #s1, we find -# # 4r 2 # #. Then (5.23) follows from (5.6) of Theorem 5.7. Note that for a Stieltjes matrix D there is a Stieltjes matrix G such that G 2 = D according to Theorem 6.15 in =-=[1]-=-. We can use G instead of D 1/2 in the proof of Corollary 5.9. Then G -1 EG -1 is a nonnegative matrix and r = r(G -1 EG -1 ) = #/2. This provides another way to prove the corollary. If we know there ... |

58 | Multisplittings of matrices and parallel solution of linear systems,” - O’Leary, White - 1985 |

56 |
Iterative methods for solving partial difference equations of elliptic type.
- Young
- 1954
(Show Context)
Citation Context ... ordered matrix. The following lemma shows the characteristic polynomial of the Jacobi matrix of A. If A is a p-constantly ordered matrix with respect to (L, U) the result was first observed by Young =-=[20]-=- for p = 2 and was extended by Varga [17] for any positive integer p. Lemma 4.2. Let A be an n × n matrix with a nonsingular diagonal D. If there exist matrices P and Q such that A is a p-consistently... |

34 |
Approximating the inverse of a matrix for use on iterative algorithms on vector processors
- Dubois, Greenbaum, et al.
- 1979
(Show Context)
Citation Context ... can be improved by iteration arithmetic. The kth power of O yields an approximate inverse M -1 k given by (3.2), which is the truncation of Neumann series and was first studied as preconditioning in =-=[2]-=-. However, M is usually not Hermitian. This brings some difficulty when M is applied as a preconditioner for a Hermitian positive definite matrix, in particular, if a preconditioned conjugate gradient... |

27 |
of values and iterative methods
- Fields
- 1993
(Show Context)
Citation Context ...Ax) : x # C n , (x, x) = 1} is called the field of values or the numerical range of A. Numerical radius is considered to be an e#cient norm to measure convergence of basic iterative methods. See [2], =-=[4]-=-, [11] and [14] for some examples. For a nonnegative matrix A, in 1975, Goldberg, Tadmor and Zwas [6] showed that the numerical radius of A is equal to the spectral radius of its symmetric part. Lemma... |

18 |
An elementary proof of the power inequality for the numerical radius,
- Pearcy
- 1966
(Show Context)
Citation Context ...y, Ay) ((S # ) -1 y, (S # ) -1 y) = (z, SAS # z) (z, z) , which implies the desired result, where y = R # x and z = (S # ) -1 y. Other properties of numerical radius can be found in [5], [8], [9] and =-=[13]-=-. A Stieltjes matrix is a symmetric M-matrix. For a Stieltjes matrix we have the following property on its Cholesky factorization. Lemma 5.4. Let A be a Stieltjes matrix and A = e L e L T be the Chole... |

17 |
Numerical determination of the field of values of a general complex matrix
- JOHNSON
- 1978
(Show Context)
Citation Context ...1 y) = (y, Ay) ((S # ) -1 y, (S # ) -1 y) = (z, SAS # z) (z, z) , which implies the desired result, where y = R # x and z = (S # ) -1 y. Other properties of numerical radius can be found in [5], [8], =-=[9]-=- and [13]. A Stieltjes matrix is a symmetric M-matrix. For a Stieltjes matrix we have the following property on its Cholesky factorization. Lemma 5.4. Let A be a Stieltjes matrix and A = e L e L T be ... |

16 |
Gauss-Seidel Methods for Solving Large Systems of Linear Equations
- Kahan
- 1958
(Show Context)
Citation Context ...S# stands for S# = (D - #P ) -1 ((1 - #)D + #Q) (3.2) with a real parameter #. Applying the results on matrices in Ln in the previous section, we have the following result similarly to Kahan's result =-=[10]-=- on the SOR method. Theorem 3.1. Let A = D - P - Q such that P, Q # Ln . If #(S# )s1, then 0s2. Proof. Represent S# = (I -#D -1 P ) -1 ((1 -#)I +#D -1 Q). Under the condition of the theorem we find #D... |

12 |
p-Cyclic Matrices: A Generalization of the Young-Frankel Scheme
- Varga
- 1959
(Show Context)
Citation Context ... extended to the generalization. The asymptotic rate of convergence of the new method is derived for Hermitian positive definite matrices. These extend some elegant results of the SOR method in Varga =-=[9]-=-, [10] and Young [11], [12]. This paper continues the study of application of stair matrices and their generalizations to iterative methods focusing on construction of convergent splittings and precon... |

11 |
Iterative Solution of Large Sparse Linear Systems of Equations
- Hackbusch
- 1994
(Show Context)
Citation Context ... have . the Jacobi method if M = D, . the Gauss-Seidel method if M = D - L and . the SOR method if M = D # - L, where # is a real parameter. Detail discussions of basic iterative methods are found in =-=[7]-=-, [18] and [22]. The Jacobi method is easily implemented on parallel computing platforms, but it is neither robust nor as fast as the Gauss-Seidel method and the SOR method in sequential case. With a ... |

10 |
iterative analysis
- Matrix
- 1962
(Show Context)
Citation Context ...ve • the Jacobi method if M = D, • the Gauss-Seidel method if M = D − L and • the SOR method if M = D ω − L, whereωis a real parameter. Detail discussions of basic iterative methods are found in [7], =-=[18]-=- and [22]. The Jacobi method is easily implemented on parallel computing platforms, but it is neither robust nor as fast as the Gauss-Seidel method and the SOR method in sequential case. With a proper... |

9 | On the numerical radius and its applications
- Goldberg, Tadmor
- 1982
(Show Context)
Citation Context ..., (R # ) -1 y) = (y, Ay) ((S # ) -1 y, (S # ) -1 y) = (z, SAS # z) (z, z) , which implies the desired result, where y = R # x and z = (S # ) -1 y. Other properties of numerical radius can be found in =-=[5]-=-, [8], [9] and [13]. A Stieltjes matrix is a symmetric M-matrix. For a Stieltjes matrix we have the following property on its Cholesky factorization. Lemma 5.4. Let A be a Stieltjes matrix and A = e L... |

8 |
On the Linear Iteration Procedures for Symmetric Matrices
- OSTROWSKI
- 1954
(Show Context)
Citation Context ... Let A = D-E-E # with a Hermitian positive definite matrix D and assume that D - #E is nonsingular for 0 # # # 2. In 1954, Ostrowski showed that #(S# )s1 if and only if A is positive definite and 0s2 =-=[12]-=-. In 1973, Varga [19] pointed out that D-#E is nonsingular for 0 # # # 2 if A and D are positive definite. Combining Varga's observation with Ostrowski's Theorem [12] (see also [18]) we immediately ob... |

6 |
On the Solution of Linear Simultaneous Equations by Iteration
- Stein, Rosenberg
- 1948
(Show Context)
Citation Context ...called an M-matrix. We now show that the important result on the SOR method for Zmatrices given by Young [22] (Theorem 5.1, pages 120--122), which is an extension of the result of Stein and Rosenberg =-=[15]-=- on the Gauss-Seidel method, is still true for the new method (3.1). Theorem 3.2. Let A be a Z-matrix with nonsingular diagonal D. Split A = D - P -Q such that P # Ln and P, Q are nonnegative matrices... |

5 |
Iterative Methods for Solving Partial Dierence Equations of Elliptic Type
- Young
- 1950
(Show Context)
Citation Context ... ordered matrix. The following lemma shows the characteristic polynomial of the Jacobi matrix of A. If A is a p-constantly ordered matrix with respect to (L, U) the result was first observed by Young =-=[20]-=- for p = 2 and was extended by Varga [17] for any positive integer p. Lemma 4.2. Let A be an n n matrix with a nonsingular diagonal D. If there exist matrices P and Q such that A is a p-consistently o... |

4 |
of values and the ADI method for nonnormal matrices
- Starke, Field
- 1993
(Show Context)
Citation Context ..., (x, x) = 1} is called the field of values or the numerical range of A. Numerical radius is considered to be an e#cient norm to measure convergence of basic iterative methods. See [2], [4], [11] and =-=[14]-=- for some examples. For a nonnegative matrix A, in 1975, Goldberg, Tadmor and Zwas [6] showed that the numerical radius of A is equal to the spectral radius of its symmetric part. Lemma 5.1. If A is a... |

3 | Numerical radius for positive matrices
- Goldberg, Tadmor, et al.
- 1975
(Show Context)
Citation Context ...us is considered to be an e#cient norm to measure convergence of basic iterative methods. See [2], [4], [11] and [14] for some examples. For a nonnegative matrix A, in 1975, Goldberg, Tadmor and Zwas =-=[6]-=- showed that the numerical radius of A is equal to the spectral radius of its symmetric part. Lemma 5.1. If A is an n n nonnegative matrix, then r(A) = #(H(A)). In general, we have the following bound... |

3 |
Matrix Anlysis
- Horn, Johnson
- 1990
(Show Context)
Citation Context ...# ) -1 y) = (y, Ay) ((S # ) -1 y, (S # ) -1 y) = (z, SAS # z) (z, z) , which implies the desired result, where y = R # x and z = (S # ) -1 y. Other properties of numerical radius can be found in [5], =-=[8]-=-, [9] and [13]. A Stieltjes matrix is a symmetric M-matrix. For a Stieltjes matrix we have the following property on its Cholesky factorization. Lemma 5.4. Let A be a Stieltjes matrix and A = e L e L ... |

2 |
Iterative Analysis, Prentice-Hall
- Matrix
- 1962
(Show Context)
Citation Context ... . the Jacobi method if M = D, . the Gauss-Seidel method if M = D - L and . the SOR method if M = D # - L, where # is a real parameter. Detail discussions of basic iterative methods are found in [7], =-=[18]-=- and [22]. The Jacobi method is easily implemented on parallel computing platforms, but it is neither robust nor as fast as the Gauss-Seidel method and the SOR method in sequential case. With a proper... |

2 |
The block symmetric successive overrelaxation method
- Ehrlich
- 1964
(Show Context)
Citation Context ...cientific Computation, Texas A&M University, College Station, Texas 77843-3404, USA (na.hlu@na-net.ornl.gov) 1 result on convergence of the SSOR method due to Habetler and Wachspress [5], and Ehrlich =-=[3]-=-, and Young [12]. Furthermore, preconditioning average is introduced to improve the approximate inverse preconditionings. However, the issue is addressed in a general framework, which can be applied t... |

1 |
On the numerical radius of matrices and its application to iteration methods, Linear and multilinear Algebra
- Axelssoni, Lu, et al.
- 1994
(Show Context)
Citation Context ...{(x, Ax) : x # C n , (x, x) = 1} is called the field of values or the numerical range of A. Numerical radius is considered to be an e#cient norm to measure convergence of basic iterative methods. See =-=[2]-=-, [4], [11] and [14] for some examples. For a nonnegative matrix A, in 1975, Goldberg, Tadmor and Zwas [6] showed that the numerical radius of A is equal to the spectral radius of its symmetric part. ... |

1 |
Forward-backward heat equations and analysis of iterative methods
- Lu
- 1995
(Show Context)
Citation Context ... x # C n , (x, x) = 1} is called the field of values or the numerical range of A. Numerical radius is considered to be an e#cient norm to measure convergence of basic iterative methods. See [2], [4], =-=[11]-=- and [14] for some examples. For a nonnegative matrix A, in 1975, Goldberg, Tadmor and Zwas [6] showed that the numerical radius of A is equal to the spectral radius of its symmetric part. Lemma 5.1. ... |

1 |
Orderings of the successive overlaxation scheme
- Varga
- 1959
(Show Context)
Citation Context ...) is the spectral radius of the associated Jacobi matrix. For p = 2, # # can be expressed equivalently as # # = 2 1 + p 1 - # 2 (B) = 1 + #(B) 1 + p 1 - # 2 (B) ! 2 . (4.8) Following Varga's approach =-=[16]-=-, we immediately obtain the following result on the optimum parameter. Theorem 4.4. Let A = D-P-Q be a p-consistently ordered matrix with respect to (P, Q), where D is a nonsingular matrix and P # Ln ... |

1 |
of the successive overrelation theory with applications of finite element approximations
- Extensions
- 1973
(Show Context)
Citation Context ... a Hermitian positive definite matrix D and assume that D - #E is nonsingular for 0 # # # 2. In 1954, Ostrowski showed that #(S# )s1 if and only if A is positive definite and 0s2 [12]. In 1973, Varga =-=[19]-=- pointed out that D-#E is nonsingular for 0 # # # 2 if A and D are positive definite. Combining Varga's observation with Ostrowski's Theorem [12] (see also [18]) we immediately obtain the following re... |

1 |
Symmetric successive overrelaxation in solving diifusion di#erence equations
- Habetler, Wachspress
- 1961
(Show Context)
Citation Context ...+ Institute for Scientific Computation, Texas A&M University, College Station, Texas 77843-3404, USA (na.hlu@na-net.ornl.gov) 1 result on convergence of the SSOR method due to Habetler and Wachspress =-=[5]-=-, and Ehrlich [3], and Young [12]. Furthermore, preconditioning average is introduced to improve the approximate inverse preconditionings. However, the issue is addressed in a general framework, which... |

1 |
An alternative approach to estimation of the conjugate gradient iteration number
- Kaporin
(Show Context)
Citation Context ... and denote (D) = # 1 n n # i=1 # i # n # n # i=1 #n . (4.3) It is readily seen that (D) = # 1 n tr(D) # n /det(D), where tr(D) is the trace of D and det(D) is the determinant of D. Following Kaporin =-=[6]-=- we illustrate the following results. The results are also found in [1]. a) Let A and B be Hermitian positive matrices then (#A + #B) # max((A), (B)), where # and # are nonnegative constants. b) Let A... |