B. N. Parlett and Y. Saad, Complex shift and invert strategies for real matrices, Linear Algebra Appl., 88-89 (1987), pp. 575--595.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the Arnoldi or unsymmetric Lanczos procedures have to be carried out in complex arithmetic. We note that each eigenvalue near oe is related to an eigenvalue near oe. To preserve the structure, we need to extract these pairs of eigenvalues together. In this case, the obvious transformation [114] is f oe ( Gamma oe) Gamma oe) The corresponding S is given by S = A Gamma oeB) 1 Im(oe) A Gamma oeB) 6.4) Note that S is now real and the eigenvectors of A Gamma B are eigenvectors of S. In practice, because of (6.3) this approach requires a complex ....

B. N. Parlett and Y. Saad, Complex shift and invert strategies for real matrices, Linear Algebra Appl., 88-89 (1987), pp. 575--595.

....An LU factorization of Q(oe) immediately yields factorizations of Q(oe) Q( Gammaoe) and Q( Gammaoe) Thus one sparse LU factorization is all we need. There are several other ways to apply (W Gamma oeI) eOEciently to a real vector for real W and complex oe. One possibility is described in [30]. Another possibility is to embed the matrix in a double sized real matrix and then use a real factorization. Which of these dioeerent approaches is best will depend on the structure of the matrices G, K, amd M and also on the choice of shifts. To derive the formulas for applying the operator R 1 ....

....the following table. a sparse LU factorization of Q( 0 ) is available. 9 saxpy operations. In general the arithmetic is complex; however, if 0 is real, then all operations are real. If 0 is neither real nor purely imaginary, there is a trick that halves the cost of computing R 1 ( 0 ; W) [30]. In this case one easily veries that R 1 ( 0 ; W) is proportional to the imaginary part of R 2 ( 0 ; W) so we can obtain R 1 ( 0 ; W) at the costs listed in Table 1 (with complex arithmetic) by simply computing R 2 ( 0 ; W) and keeping the imaginary part. The symplectic operators S 1 and S 2 ....

B.N. Parlett and Y. Saad. Complex shift and invert strategies for real matrices. Linear Algebra Appl., 88/89:575595, 1987.

....factorization of Q(oe) immediately yields factorizations of Q(oe) Q( Gammaoe) and Q( Gammaoe) Thus one sparse LU factorization is all we need. There are several other ways to apply (W Gamma oeI) Gamma1 eOEciently to a real vector for real W and complex oe. One possibility is described in [30]. Another possibility is to embed the matrix in a double sized real matrix and then use a real factorization. Which of these dioeerent approaches is best will depend on the structure of the matrices G, K, amd M and also on the choice of shifts. To derive the formulas for applying the operator R 1 ....

....Matrix vector products Mz 5 skew symmetric sparse Matrix vector products Gz 9 saxpy operations. In general the arithmetic is complex; however, if 0 is real, then all operations are real. If 0 is neither real nor purely imaginary, there is a trick that halves the cost of computing R 1 ( 0 ; W) [30]. In this case one easily veries that R 1 ( 0 ; W) is proportional to the imaginary part of R 2 ( 0 ; W) so we can obtain R 1 ( 0 ; W) at the costs listed in Table 1 (with complex arithmetic) by simply computing R 2 ( 0 ; W) and keeping the imaginary part. The symplectic operators S 1 and S 2 can ....

B.N. Parlett and Y. Saad. Complex shift and invert strategies for real matrices. Linear Algebra Appl., 88/89:575595, 1987.

....the Arnoldi or unsymmetric Lanczos procedures have to be carried out in complex arithmetic. We note that each eigenvalue near oe is related to an eigenvalue near oe. To preserve the structure, we need to extract these pairs of eigenvalues together. In this case, the obvious transformation [114] is f oe ( Gamma oe) Gamma1 ( Gamma oe) Gamma1 . The corresponding S is given by S = A Gamma oeB) Gamma1 B(A Gamma oeB) Gamma1 B = 1 Im(oe) Im Gamma (A Gamma oeB) Gamma1 B Delta : 6.4) Note that S is now real and the eigenvectors of A Gamma B are ....

B. N. Parlett and Y. Saad, Complex shift and invert strategies for real matrices, Linear Algebra Appl., 88-89 (1987), pp. 575--595.

....the Arnoldi or unsymmetric Lanczos procedures have to be carried out in complex arithmetic. We note that each eigenvalue near oe is related to an eigenvalue near oe. To preserve the structure, we need to extract these pairs of eigenvalues together. In this case, the obvious transformation [104] is f oe ( Gamma oe) Gamma1 ( Gamma oe) Gamma1 . The corresponding S is given by S = A Gamma oeB) Gamma1 B(A Gamma oeB) Gamma1 B = 1 Im(oe) Im Gamma (A Gamma oeB) Gamma1 B Delta : 6.4) Note that S is now real and has the same eigenvectors as (A; B) In ....

Beresford N. Parlett and Youcef Saad, Complex shift and invert strategies for real matrices, Linear Algebra and Appl., 88/89 (1987), pp. 575--595.

....at the start of the computation. If B is singular, the problem of solving (3.2) becomes harder. A shift g may be introduced so that (A g B) is nonsingular and then (A g Bx) l g) Bx (3.3) may be treated as a standard eigenvalue problem by working with the matrix 1 (A g B) B. 3. 4) Parlett and Saad (1985) have considered ways of dealing with the case where A and B are real but the shift g is complex. One possibility is to replace the matrix (3.4) by the real matrix 1 Re[ A g B) B] 3.5) which has the same eigenvectors as the original problem and eigenvalues n which are related to the i ....

Parlett, B. N. and Saad, Y. (1985). Complex shift and invert strategies for real matrices. Technical Report YALEU/DCS-RR-424, Yale University Department of Computer Science.

....in the second [14] we show how to treat cases when both A and B may be singular, and how to apply the perturbation theory for regular pencils from [18] to bound the errors. In this third report, we show how to apply complex double shifts to a real pencil (1) extending an idea of Parlett and Saad [8]. We will continue in section 2, by reviewing the basic recursion that gives an orthogonal basis V , where the pencil (A; B) 1) is represented by a Hessenberg pencil (K; H) In section 3, we show how to get Ritz approximations to eigenvalues and eigenvectors from this Hessenberg pencil. In ....

B. Parlett and Y. Saad, Complex shift and invert strategies for real matrices, Lin. Alg. Appl., 88/89 (1987), pp. 575--595.

....the sparse matrix option in Matlab4. Reorthogonalization was done in step 3 whenever necessary. We took advantage of the complex arithmetic in Matlab, even when we had real matrices. It is straightforward, but not entirely simple to make a program that preserves reality in the way indicated in [6], but we have not yet taken the effort to do it. Problems of sizes n up to 2000 and runs up to j = 40 have been handled. Let us report results from a hydrodynamical bifurcation computation that has been treated extensively in the literature, see [5] 4] and [3] It is a parametrized nonlinear ....

B. Parlett and Y. Saad, Complex shift and invert strategies for real matrices, Lin. Alg. Appl., 88/89 (1987), pp. 575--595.

....i fi j ) Proof. The proof is given in [13] using the Alternative Theorem [12, 11] 2 Shift invert will rapidly converge to any eigenvalue in T . The complex arithmetic to compute the transformations and perform subspace iteration can be reduced partly by using techniques from Parlett and Saad [15]. However, these are not considered here because they change the transformations in a rather complicated way or destroy possibly advantageous sparsity in the applications we have in mind. The iterations are stopped when ffl i ffl tol ; i = 1 : r but if T does not contain eigenvalues, ....

B.N. Parlett and Y. Saad. Complex Shift and Invert Strategies for Real Matrices. Technical Report YALEU/DCS/RR-424, University of California at Berkeley, Mathematics Department, 1985.

.... starting vectors is not deficient in the eigenvector associated with the desired eigenvalue [1, p208 209] In many applications the shift will be complex and this means working in complex arithmetic, or doubling the size of the system to work in real arithmetic, or using some other strategy (see [11]) The goal of this paper is to present a real transformation for use when oe is complex which has, roughly speaking, the same desirable mapping properties as T si and which has some advantages in stability applications, but which allows the use of real arithmetic and uses real systems of size N ....

B.N. Parlett and Y. Saad. Complex Shift and Invert Strategies for Real Matrices. Linear Algebra Appl., 88/89:575--595, 1987.

....interior eigenvalues. Furthermore, this shift and invert strategy may lead to a better separation of the desired eigenspectrum, which can result in faster convergence. The commonly followed procedure for solving generalized non hermitian eigenvalue problems is the shift and invert Arnoldi method [14, 17]. Recently developed variants of this technique involve multiple shifts in one run [16] Another option is the generalized non hermitian Lanczos procedure [4] which is also based upon a shift and invert strategy. When the order of the matrices becomes so large that factorization is very ....

B.N. Parlett and Y. Saad, Complex shift and invert strategies for real matrices, Linear Algebra Appl., 88/89 (1987), pp. 575--595.

....generated and applied to G such that bulges introduced by the previous reflector is pushed towards the upper left corner of H . We summarize the above discussion in Algorithm 5. We remark that the matrix A is not generally formed. We may apply some of the techniques discussed in Parlett and Saad [16] when using direct factorizations, or perform separate successive matrix vector products when using an iterative method. Algorithm 5: DBTRQ) Doubly shifted Truncated RQ iteration Input: A; V k 1 ; H k 1 ; f k 1 ) with AV k 1 = V k 1 H k 1 f k 1 e T k 1 ; V H k 1 V k 1 = I , H k upper ....

B. N. Parlett and Y. Saad. Complex shift and invert strategies for real matrices. Linear Algebra and its Applications, 88/89:575--595, 1987.

....while the 5th, and 6th degree polynomials are deficient. After 8 steps, the algorithm finds invariant subspaces with respect to both A and A T , and the computed eigenvalues match the true eigenvalues exactly, as shown in Figure 1. Example 2. This example is an eigenvalue problem, taken from [12], whose exact eigenvalues are known. Generally, problems of this type arise in modeling concentration waves in reaction and transport interaction of chemical solutions in a tubular reactor. The particular test problem used here corresponds to the so called Brusselator wave model. We took N = 200 ....

Parlett, B. N., and Saad, Y. Complex shift and invert strategies for real matrices. Linear Algebra Appl. 88/89 (1987), 575--595.

....A. In many cases, the standard and the look ahead Lanczos procedures give similar results. In particular, for this example, the results obtained from both the standard and the look ahead Lanczos algorithm match those reported in [6] Example 6:2. This example is an eigenvalue problem, taken from [24], whose exact eigenvalues are known. Generally, problems of this type arise in modeling concentration waves in reaction and transport interaction of chemical solutions in a tubular reactor. The particular test problem used here corresponds to the so called Brusselator wave model. This example was ....

B.N. Parlett and Y. Saad, Complex shift and invert strategies for real matrices, Linear Algebra Appl., 88/89 (1987), pp. 575--595.

....oeM = LU , and then solving the linear systems of equations LUY = MX and Z T = X T (LU ) Gamma1 M for Y and Z T , respectively. If K and M are real, and the shift oe is complex, one can still keep the Lanczos procedure in real arithmetic using a strategy proposed by Parlett and Saad [37]. In many applications, M is symmetric positive definite. In this case, one can avoid factoring M explicitly by preserving M biorthogonality among the Lanczos vectors [16, 35, 11, 22] Numerical methods for the case in which M is symmetric indefinite are discussed in [22, 32] 7. Summary of ....

B. Parlett and Y. Saad. Complex shift and invert strategies for real matrices. Lin. Alg. Appl., 88/89:575--595, 1987.

....constants in a typical example is 40 fold faster than in the old version. The reduction of the memory consumption amounts the 20 fold in the represented example, since the sparse storage technique is applied. We use the nonsymmetric version of the implicitly restarted Arnoldi iteration [17] [15], 13] 18] 11] The Arnoldi iteration produces a partial orthogonal factorization of a matrix A of order m into an upper Hessenberg matrix H r of order r, r m. Using the eigenvalues of this small matrix an approximation of a subset of the eigenvalues of the matrix A can be obtained. The ....

Parlett, B. N., Saad, Y., Complex Shift and Invert Strategies for Real Matrices, Linear Algebra and its Applications, vol 88/89, pp. 575-595, 1987.

....stored as a dense matrix. In the following we show that one can avoid the computation of all eigenvalues to find the few required propagation constants using an iterative method which has to be carried out twice. We use the nonsymmetric version of the implicitly restarted Arnoldi iteration [9] [10], 11] 12] The standard eigenvalue problem Ax = x can be solved using the regular mode Ax = x or the inverse mode A Gamma1 x = 1 x. The Arnoldi algorithm is applied iteratively to solve one of these problems generating Arnoldi vectors. Using the regular mode most of the cost in ....

Parlett, B.N., Saad, Y., Complex Shift and Invert Strategies for Real Matrices, Linear Algebra and its Applications, Vol. 88/89, 1987, pp. 575-595.

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B. N. Parlett and Y. Saad. Complex shift and invert strategies for real matrices. Linear Algebra and its Applications, 88/89:575-595, 1987.

....Shift and invert has now become a fairly standard tool in structural analysis because of the predominance of generalized eigenvalue problems in this applications area. For nonsymmetric eigenvalue problems, much remains to be done to derive efficient Shift and Invert strategies. Parlett and Saad [24] have examined different ways of dealing with the situation where the matrices M and K are real and banded but the shift rr is complex. One such possibility is to replace the complex operator (K rrM) iM by the real one Re[ K o M) iM] 31) whose eigenvectors are the same as those of the original ....

B. N. Parlett and Y. Saad. Complex shift and invert strategies for real matrices. Linear Algebra and its Applications, 88/89:575-595, 1987.

....are sought. The matrix (A Gamma oeI) Gamma1 , need not be explicitly computed: all we need is to factor (A Gamma oeI) into LU and subsequently at each step of the iterative method solve two triangular systems one with L and the other with U . Thus band structure can be fully exploited. In [18] several implementations of the shift and invert strategy are considered and the problem of avoiding complex arithmetic when A is real is addressed. An important application of the eigenvalue algorithms is to provide a small invariant subspace that will represent the critical modes of the system. ....

B. N. Parlett and Y. Saad. Complex shift and invert strategies for real matrices. Linear Algebra and its Applications, 88/89:575--595, 1987.

....reduced matrix pencil (T m ; Dm ) are computed, The approximate eigentriplets of A are selected from the Ritz triplets f( i ; Qm z i ; Pmw i )g. Transformation functions Operation References polynomial p(x) p(A)v [39, 40] shifting and inverting (x Gamma ) Gamma1 (A Gamma ) Gamma1 v [36] Cayley transformation c(x) x Gamma ff) x ff) A ffI) Gamma1 (A Gamma ffI)v [7, 11, 22] rational function r(x) p(x) q(x) r(A)v [37, 38] exponential function e x e A v p(A)v r(A)v [19, 11, 43] Table 1: Spectrum Transformation A user should note that for simultaneous iteration ....

....schur vectors closest to the shift oe of a banded matrix A. The matrix A is passed in banded format and then A Gamma oeI is factorized by LINPACK which is then used as the operator for Arnoldi method. Although this operator may be complex, the main computations are done in real arithmetic [36]. User can decide to use a new shift after a certain number of restarted Arnoldi when computing some eigenvalue. 3. ARNUPD by Sorensen [47] treats the residual vector, after a run of Arnoldi process, as a function of the initial vector. This initial vector is then updated through a chosen ....

B. Parlett and Y. Saad. Complex shift and invert strategies for real matrices. Lin. Alg. and Appl., 88/89:575--595, 1987.

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B. N. Parlett and Y. Saad, Complex shift and invert strategies for real matrices, Linear Algebra Appl., 88-89 (1987), pp. 575--595.

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B.N. Parlett and Y. Saad. Complex shift and invert strategies for real matrices. Linear Algebra and Its Applications, 88/89(1), 575--595, 1987.

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REFERENCES 41 B.N. Parlett and Y. Saad. Complex shift and invert strategies for real matrices. Linear Algebra and Its Applications, 88/89(1), 575--595, 1987.

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