Post on 21-Oct-2020
BasicsPower Iteration
QR IterationKrylov MethodsOther Methods
Parallel Numerical AlgorithmsChapter 5 – Eigenvalue Problems
Section 5.2 – Eigenvalue Computation
Michael T. Heath and Edgar Solomonik
Department of Computer ScienceUniversity of Illinois at Urbana-Champaign
CS 554 / CSE 512
Michael T. Heath and Edgar Solomonik Parallel Numerical Algorithms 1 / 42
BasicsPower Iteration
QR IterationKrylov MethodsOther Methods
Outline
1 Basics
2 Power Iteration
3 QR Iteration
4 Krylov Methods
5 Other Methods
Michael T. Heath and Edgar Solomonik Parallel Numerical Algorithms 2 / 42
BasicsPower Iteration
QR IterationKrylov MethodsOther Methods
DefinitionsTransformationsParallel Algorithms
Eigenvalues and Eigenvectors
Given n× n matrix A, find scalar λ and nonzero vector xsuch that
Ax = λx
λ is eigenvalue and x is corresponding eigenvector
A always has n eigenvalues, but they may be neither realnor distinct
May need to compute only one or few eigenvalues, or all neigenvalues
May or may not need corresponding eigenvectors
Michael T. Heath and Edgar Solomonik Parallel Numerical Algorithms 3 / 42
BasicsPower Iteration
QR IterationKrylov MethodsOther Methods
DefinitionsTransformationsParallel Algorithms
Problem Transformations
Shift : for scalar σ, eigenvalues of A− σI are eigenvaluesof A shifted by σ, λi − σ
Inversion : for nonsingular A, eigenvalues of A−1 arereciprocals of eigenvalues of A, 1/λi
Powers : for integer k > 0, eigenvalues of Ak are kthpowers of eigenvalues of A, λki
Polynomial : for polynomial p(t), eigenvalues of p(A) arevalues of p evaluated at eigenvalues of A, p(λi)
Michael T. Heath and Edgar Solomonik Parallel Numerical Algorithms 4 / 42
BasicsPower Iteration
QR IterationKrylov MethodsOther Methods
DefinitionsTransformationsParallel Algorithms
Similarity Transformations
B is similar to A if there is nonsingular T such that
B = T−1AT
Then
By = λy ⇒ T−1ATy = λy ⇒ A(Ty) = λ(Ty)
so A and B have same eigenvalues, and if y iseigenvector of B, then x = Ty is eigenvector of A
Similarity transformations preserve eigenvalues, andeigenvectors are easily recovered
Michael T. Heath and Edgar Solomonik Parallel Numerical Algorithms 5 / 42
BasicsPower Iteration
QR IterationKrylov MethodsOther Methods
DefinitionsTransformationsParallel Algorithms
Similarity Transformations
Forms attainable by similarity transformation
A T Bdistinct eigenvalues nonsingular diagonalreal symmetric orthogonal real diagonalcomplex Hermitian unitary real diagonalnormal unitary diagonalarbitrary real orthogonal real block triangular (real Schur)arbitrary unitary upper triangular (Schur)arbitrary nonsingular almost diagonal (Jordan)
Michael T. Heath and Edgar Solomonik Parallel Numerical Algorithms 6 / 42
BasicsPower Iteration
QR IterationKrylov MethodsOther Methods
DefinitionsTransformationsParallel Algorithms
Preliminary Reduction
Eigenvalues easier to compute if matrix first reduced tosimpler form by similarity transformation
Diagonal or triangular most desirable, but cannot alwaysbe reached in finite number of steps
Preliminary reduction usually to tridiagonal form (forsymmetric matrix) or Hessenberg form (for nonsymmetricmatrix)
Preliminary reduction usually done by orthogonaltransformations, using algorithms similar to QRfactorization, but transformations must be applied fromboth sides to maintain similarity
Michael T. Heath and Edgar Solomonik Parallel Numerical Algorithms 7 / 42
BasicsPower Iteration
QR IterationKrylov MethodsOther Methods
DefinitionsTransformationsParallel Algorithms
Parallel Algorithms for Eigenvalues
Algorithms for computing eigenvalues and eigenvectorsemploy basic operations such as
vector updates (saxpy)inner productsmatrix-vector and matrix-matrix multiplicationsolution of triangular systemsorthogonal (QR) factorization
In many cases, parallel implementations will be based onparallel algorithms we have already seen for these basicoperations, although there will sometimes be new sourcesof parallelism
Michael T. Heath and Edgar Solomonik Parallel Numerical Algorithms 8 / 42
BasicsPower Iteration
QR IterationKrylov MethodsOther Methods
Power IterationInverse IterationSimultaneous Iteration
Power Iteration
Simplest method for computing one eigenvalue-eigenvector pair is power iteration, which in effect takessuccessively higher powers of matrix times initial startingvector
x0 = arbitrary nonzero vectorfor k = 1, 2, . . .
yk = Axk−1xk = yk/‖yk‖∞
end
If A has unique eigenvalue λ1 of maximum modulus, thenpower iteration converges to eigenvector corresponding todominant eigenvalue
Michael T. Heath and Edgar Solomonik Parallel Numerical Algorithms 9 / 42
BasicsPower Iteration
QR IterationKrylov MethodsOther Methods
Power IterationInverse IterationSimultaneous Iteration
Power Iteration
Convergence rate of power iteration depends on ratio|λ2/λ1|, where λ2 is eigenvalue having second largestmodulus
It may be possible to choose shift, A− σI, so that ratio ismore favorable and yields more rapid convergence
Shift must then be added to result to obtain eigenvalue oforiginal matrix
Michael T. Heath and Edgar Solomonik Parallel Numerical Algorithms 10 / 42
BasicsPower Iteration
QR IterationKrylov MethodsOther Methods
Power IterationInverse IterationSimultaneous Iteration
Parallel Power Iteration
Power iteration requires repeated matrix-vector products,which are easily implemented in parallel for dense orsparse matrix, as we have seen
Additional communication may be required fornormalization, shifts, convergence test, etc.
Though easily parallelized, power iteration is often too slowto be useful in this form
Michael T. Heath and Edgar Solomonik Parallel Numerical Algorithms 11 / 42
BasicsPower Iteration
QR IterationKrylov MethodsOther Methods
Power IterationInverse IterationSimultaneous Iteration
Inverse Iteration
Inverse iteration is power iteration applied to A−1, whichconverges to eigenvector corresponding to dominanteigenvalue of A−1, which is reciprocal of smallesteigenvalue of A
Inverse of A is not computed explicitly, but onlyfactorization of A (and only once) to solve system of linearequations at each iteration
x0 = arbitrary nonzero vectorfor k = 1, 2, . . .
Solve Ayk = xk−1 for ykxk = yk/‖yk‖∞
end
Michael T. Heath and Edgar Solomonik Parallel Numerical Algorithms 12 / 42
BasicsPower Iteration
QR IterationKrylov MethodsOther Methods
Power IterationInverse IterationSimultaneous Iteration
Inverse Iteration
Shifting strategy can greatly accelerate convergence
Inverse iteration is especially useful for computingeigenvector corresponding to approximate eigenvalue,since it converges rapidly when approximate eigenvalue isused as shift
Inverse iteration also useful for computing eigenvalueclosest to any given value β, since if β is used as shift,then desired eigenvalue corresponds to smallesteigenvalue of shifted matrix
Michael T. Heath and Edgar Solomonik Parallel Numerical Algorithms 13 / 42
BasicsPower Iteration
QR IterationKrylov MethodsOther Methods
Power IterationInverse IterationSimultaneous Iteration
Parallel Inverse Iteration
Inverse iteration requires initial factorization of matrix Aand solution of triangular systems at each iteration, so itappears to be much less amenable to efficient parallelimplementation than power iteration
However, inverse iteration is often used to computeeigenvector in situations where
approximate eigenvalue is already known, so using it asshift yields very rapid convergencematrix has previously been reduced to simpler form (e.g.,tridiagonal) for which linear system is easy to solve
Michael T. Heath and Edgar Solomonik Parallel Numerical Algorithms 14 / 42
BasicsPower Iteration
QR IterationKrylov MethodsOther Methods
Power IterationInverse IterationSimultaneous Iteration
Simultaneous Iteration
To compute many eigenvalue-eigenvector pairs, couldapply power iteration to several starting vectorssimultaneously, giving simultaneous iteration
X0 = arbitrary n× q matrix of rank qfor k = 1, 2, . . .
Xk = AXk−1end
span(Xk) converges to invariant subspace determined byq largest eigenvalues of A, provided |λq| > |λq+1|
Normalization is needed at each iteration, and columns ofXk become increasingly ill-conditioned basis for span(Xk)
Michael T. Heath and Edgar Solomonik Parallel Numerical Algorithms 15 / 42
BasicsPower Iteration
QR IterationKrylov MethodsOther Methods
Orthogonal IterationQR Iteration
Orthogonal Iteration
Both issues addressed by computing QR factorization ateach iteration, giving orthogonal iteration
X0 = arbitrary n× q matrix of rank qfor k = 1, 2, . . .
Compute reduced QR factorizationQ̂kRk = Xk−1
Xk = AQ̂kend
Converges to block triangular form, and blocks aretriangular where moduli of consecutive eigenvalues aredistinct
Michael T. Heath and Edgar Solomonik Parallel Numerical Algorithms 16 / 42
BasicsPower Iteration
QR IterationKrylov MethodsOther Methods
Orthogonal IterationQR Iteration
Parallel Orthogonal Iteration
Orthogonal iteration requires matrix-matrix multiplicationand QR factorization at each iteration, both of which weknow how to implement in parallel with reasonableefficiency
Michael T. Heath and Edgar Solomonik Parallel Numerical Algorithms 17 / 42
BasicsPower Iteration
QR IterationKrylov MethodsOther Methods
Orthogonal IterationQR Iteration
QR Iteration
If we take X0 = I, then orthogonal iteration shouldproduce all eigenvalues and eigenvectors of A
Orthogonal iteration can be reorganized to avoid explicitformation and factorization of matrices XkInstead, sequence of unitarily similar matrices is generatedby computing QR factorization at each iteration and thenforming reverse product, giving QR iteration
A0 = Afor k = 1, 2, . . .
Compute QR factorizationQkRk = Ak−1
Ak = RkQkend
Michael T. Heath and Edgar Solomonik Parallel Numerical Algorithms 18 / 42
BasicsPower Iteration
QR IterationKrylov MethodsOther Methods
Orthogonal IterationQR Iteration
QR Iteration
In simple form just given, each iteration of QR methodrequires Θ(n3) work
Work per iteration is reduced to Θ(n2) if matrix is inHessenberg form, or Θ(n) if symmetric matrix is intridiagonal form
Preliminary reduction is usually done by Householder orGivens transformations
In addition, number of iterations required is reduced bypreliminary reduction of matrix
Convergence rate also enhanced by judicious choice ofshifts
Michael T. Heath and Edgar Solomonik Parallel Numerical Algorithms 19 / 42
BasicsPower Iteration
QR IterationKrylov MethodsOther Methods
Orthogonal IterationQR Iteration
Parallel QR Iteration
Preliminary reduction can be implemented efficiently inparallel, using algorithms analogous to parallel QRfactorization for dense matrix
But subsequent QR iteration for reduced matrix isinherently serial, and permits little parallel speedup for thisportion of algorithm
This may not be of great concern if iterative phase isrelatively small portion of total time, but it does limitefficiency and scalability
Michael T. Heath and Edgar Solomonik Parallel Numerical Algorithms 20 / 42
BasicsPower Iteration
QR IterationKrylov MethodsOther Methods
Orthogonal IterationQR Iteration
Reduction to Hessenberg/Tridiagonal
To reduce to Hessenberg can execute QR on each column
If Hi is the Householder transformation used to annihilateith subcolumn, perform similarity transformation
A(i+1) = HiA(i)HTi
More generally, run QR on b subcolumns of A to reduce toband-width b
B(i+1) = QiB(i)QTi
To avoid fill QTi must not operate on the b columns whichQi reduces
Once reduction completed to a band-width subsequenteliminations introduce fill (bulges) but can be done
Michael T. Heath and Edgar Solomonik Parallel Numerical Algorithms 21 / 42
BasicsPower Iteration
QR IterationKrylov MethodsOther Methods
Orthogonal IterationQR Iteration
Reduction from Banded to Tridiagonal
Two-sided successive orthogonal reductions generatebulges of fill when applied to a banded matrixBulges can be chased with pipelined parallelism
Michael T. Heath and Edgar Solomonik Parallel Numerical Algorithms 22 / 42
BasicsPower Iteration
QR IterationKrylov MethodsOther Methods
Krylov SubspacesArnoldi IterationLanczos IterationKrylov Subspace Methods
Krylov Subspace Methods
Krylov subspace methods reduce matrix to Hessenberg(or tridiagonal) form using only matrix-vector multiplication
For arbitrary starting vector x0, if
Kk =[x0 Ax0 · · · Ak−1x0
]then
K−1n AKn = Cn
where Cn is upper Hessenberg (in fact, companion matrix)
Michael T. Heath and Edgar Solomonik Parallel Numerical Algorithms 23 / 42
BasicsPower Iteration
QR IterationKrylov MethodsOther Methods
Krylov SubspacesArnoldi IterationLanczos IterationKrylov Subspace Methods
Krylov Subspace Methods
To obtain better conditioned basis for span(Kn), computeQR factorization
QnRn = Kn
so thatQHn AQn = RnCnR
−1n ≡H
with H upper Hessenberg
Michael T. Heath and Edgar Solomonik Parallel Numerical Algorithms 24 / 42
BasicsPower Iteration
QR IterationKrylov MethodsOther Methods
Krylov SubspacesArnoldi IterationLanczos IterationKrylov Subspace Methods
Krylov Subspace Methods
Equating kth columns on each side of equationAQn = QnH yields recurrence
Aqk = h1kq1 + · · ·+ hkkqk + hk+1,kqk+1
relating qk+1 to preceding vectors q1, . . . , qk
Premultiplying by qHj and using orthonormality,
hjk = qHj Aqk, j = 1, . . . , k
These relationships yield Arnoldi iteration, which producesupper Hessenberg matrix column by column using onlymatrix-vector multiplication by A and inner products ofvectors
Michael T. Heath and Edgar Solomonik Parallel Numerical Algorithms 25 / 42
BasicsPower Iteration
QR IterationKrylov MethodsOther Methods
Krylov SubspacesArnoldi IterationLanczos IterationKrylov Subspace Methods
Arnoldi Iteration
x0 = arbitrary nonzero starting vectorq1 = x0/‖x0‖2for k = 1, 2, . . .
uk = Aqkfor j = 1 to k
hjk = qHj uk
uk = uk − hjkqjendhk+1,k = ‖uk‖2if hk+1,k = 0 then stopqk+1 = uk/hk+1,k
end
Michael T. Heath and Edgar Solomonik Parallel Numerical Algorithms 26 / 42
BasicsPower Iteration
QR IterationKrylov MethodsOther Methods
Krylov SubspacesArnoldi IterationLanczos IterationKrylov Subspace Methods
Arnoldi Iteration
IfQk =
[q1 · · · qk
],
thenHk = Q
Hk AQk
is upper Hessenberg matrix
Eigenvalues of Hk, called Ritz values, are approximateeigenvalues of A, and Ritz vectors given by Qky, where yis eigenvector of Hk, are corresponding approximateeigenvectors of A
Eigenvalues of Hk must be computed by another method,such as QR iteration, but this is easier problem if k � n
Michael T. Heath and Edgar Solomonik Parallel Numerical Algorithms 27 / 42
BasicsPower Iteration
QR IterationKrylov MethodsOther Methods
Krylov SubspacesArnoldi IterationLanczos IterationKrylov Subspace Methods
Arnoldi Iteration
Arnoldi iteration expensive in work and storage becauseeach new vector qk must be orthogonalized against allprevious columns of Qk, which must be stored
So Arnoldi process usually restarted periodically withcarefully chosen starting vector
Ritz values and vectors produced are often goodapproximations to eigenvalues and eigenvectors of A afterrelatively few iterations
Work and storage costs drop dramatically if matrixsymmetric or Hermitian, since recurrence then has onlythree terms and Hk is tridiagonal (so usually denoted Tk),yielding Lanczos iteration
Michael T. Heath and Edgar Solomonik Parallel Numerical Algorithms 28 / 42
BasicsPower Iteration
QR IterationKrylov MethodsOther Methods
Krylov SubspacesArnoldi IterationLanczos IterationKrylov Subspace Methods
Lanczos Iteration
q0 = 0β0 = 0x0 = arbitrary nonzero starting vectorq1 = x0/‖x0‖2for k = 1, 2, . . .
uk = Aqkαk = q
Hk uk
uk = uk − βk−1qk−1 − αkqkβk = ‖uk‖2if βk = 0 then stopqk+1 = uk/βk
end
Michael T. Heath and Edgar Solomonik Parallel Numerical Algorithms 29 / 42
BasicsPower Iteration
QR IterationKrylov MethodsOther Methods
Krylov SubspacesArnoldi IterationLanczos IterationKrylov Subspace Methods
Lanczos Iteration
αk and βk are diagonal and subdiagonal entries ofsymmetric tridiagonal matrix Tk
As with Arnoldi, Lanczos iteration does not produceeigenvalues and eigenvectors directly, but only tridiagonalmatrix Tk, whose eigenvalues and eigenvectors must becomputed by another method to obtain Ritz values andvectors
If βk = 0, then algorithm appears to break down, but in thatcase invariant subspace has already been identified (i.e.,Ritz values and vectors are already exact at that point)
Michael T. Heath and Edgar Solomonik Parallel Numerical Algorithms 30 / 42
BasicsPower Iteration
QR IterationKrylov MethodsOther Methods
Krylov SubspacesArnoldi IterationLanczos IterationKrylov Subspace Methods
Lanczos Iteration
In principle, if Lanczos algorithm is run until k = n,resulting tridiagonal matrix is orthogonally similar to A
In practice, rounding error causes loss of orthogonality,invalidating this expectation
Problem can be overcome by reorthogonalizing vectors asneeded, but expense can be substantial
Alternatively, can ignore problem, in which case algorithmstill produces good eigenvalue approximations, but multiplecopies of some eigenvalues may be generated
Michael T. Heath and Edgar Solomonik Parallel Numerical Algorithms 31 / 42
BasicsPower Iteration
QR IterationKrylov MethodsOther Methods
Krylov SubspacesArnoldi IterationLanczos IterationKrylov Subspace Methods
Krylov Subspace Methods
Virtue of Arnoldi and Lanczos iterations is ability to producegood approximations to extreme eigenvalues for k � n
Moreover, they require only one matrix-vector multiplicationby A per step and little auxiliary storage, so are ideallysuited to large sparse matrices
If eigenvalues are needed in middle of spectrum, say nearσ, then algorithm can be applied to matrix (A− σI)−1,assuming it is practical to solve systems of form(A− σI)x = y
Michael T. Heath and Edgar Solomonik Parallel Numerical Algorithms 32 / 42
BasicsPower Iteration
QR IterationKrylov MethodsOther Methods
Krylov SubspacesArnoldi IterationLanczos IterationKrylov Subspace Methods
Parallel Krylov Subspace Methods
Krylov subspace methods composed of
vector updates (saxpy)inner productsmatrix-vector multiplicationcomputing eigenvalues/eigenvectors of tridiagonal matrices
Parallel implementation requires implementing each ofthese in parallel as before
For early iterations, Hessenberg or tridiagonal matricesgenerated are too small to benefit from parallelimplementation, but Ritz values and vectors need not becomputed until later
Michael T. Heath and Edgar Solomonik Parallel Numerical Algorithms 33 / 42
BasicsPower Iteration
QR IterationKrylov MethodsOther Methods
Jacobi MethodBisection MethodsDivide-and-Conquer
Jacobi Method
Jacobi method for symmetrix matrix starts with A0 = Aand computes sequence
Ak+1 = JTk AkJk
where Jk is plane rotation that annihilates symmetric pairof off-diagonal entries in Ak
Plane rotations are applied repeatedly from both sides insystematic sweeps through matrix until magnitudes of alloff-diagonal entries are reduced below tolerance
Resulting diagonal matrix is orthogonally similar to originalmatrix, so diagonal entries are eigenvalues, andeigenvectors given by product of plane rotations
Michael T. Heath and Edgar Solomonik Parallel Numerical Algorithms 34 / 42
BasicsPower Iteration
QR IterationKrylov MethodsOther Methods
Jacobi MethodBisection MethodsDivide-and-Conquer
Parallel Jacobi Method
Jacobi method, though slower than QR iteration serially,parallelizes better
Parallel implementation of Jacobi method performs manyannihilations simultaneously, at locations chosen so thatrotations do not interfere with each other (analogous toparallel Givens QR factorization)
Computations are still rather fine grained, however, soeffectiveness is limited on parallel computers withunfavorable ratio of communication to computation speed
Michael T. Heath and Edgar Solomonik Parallel Numerical Algorithms 35 / 42
BasicsPower Iteration
QR IterationKrylov MethodsOther Methods
Jacobi MethodBisection MethodsDivide-and-Conquer
Bisection or Spectrum-Slicing
For real symmetric matrix, can determine how manyeigenvalues are less than given real number σ
By systematically choosing various values for σ (slicingspectrum at σ) and monitoring resulting count, anyeigenvalue can be isolated as accurately as desired
For example, by computing inertia using A = LDLT
factorization of A− σI for various σ, individual eigenvaluescan be isolated as accurately as desired using intervalbisection technique
Michael T. Heath and Edgar Solomonik Parallel Numerical Algorithms 36 / 42
BasicsPower Iteration
QR IterationKrylov MethodsOther Methods
Jacobi MethodBisection MethodsDivide-and-Conquer
Sturm Sequence
Another spectrum-slicing method for computing individualeigenvalues is based on Sturm sequence property ofsymmetric matrices
Let pr(σ) denote determinant of leading principal minor ofA− σI of order r
Zeros of pr(σ) strictly separate those of pr−1(σ), andnumber of agreements in sign of successive members ofsequence pr(σ), for r = 1, . . . , n, gives number ofeigenvalues of A strictly greater than σ
Determinants pr(σ) easy to compute if A transformed totridiagonal form before applying Sturm sequence technique
Michael T. Heath and Edgar Solomonik Parallel Numerical Algorithms 37 / 42
BasicsPower Iteration
QR IterationKrylov MethodsOther Methods
Jacobi MethodBisection MethodsDivide-and-Conquer
Parallel Multisection
Spectrum-slicing methods can be implemented in parallelby assigning disjoint intervals of real line to different tasks,and then each task carries out bisection process usingserial spectrum-slicing method for its subinterval
Both for efficiency of spectrum-slicing technique and forstorage scalability, matrix is first reduced (in parallel) totridiagonal form, so that each task can store entire(reduced) matrix
Load imbalance is potential problem, since differentsubintervals may contain different numbers of eigenvalues
Michael T. Heath and Edgar Solomonik Parallel Numerical Algorithms 38 / 42
BasicsPower Iteration
QR IterationKrylov MethodsOther Methods
Jacobi MethodBisection MethodsDivide-and-Conquer
Parallel Multisection
Clustering of eigenvalues cannot be anticipated inadvance, so dynamic load balancing may be required toachieve reasonable efficiency
Eigenvectors must still be determined, usually by inverseiteration
Since each task holds entire (tridiagonal) matrix, each canin principle carry out inverse iteration for its eigenvalueswithout any communication among tasks
Orthogonality of resulting eigenvectors cannot beguaranteed, however, and thus communication is requiredto orthogonalize them
Michael T. Heath and Edgar Solomonik Parallel Numerical Algorithms 39 / 42
BasicsPower Iteration
QR IterationKrylov MethodsOther Methods
Jacobi MethodBisection MethodsDivide-and-Conquer
Divide-and-Conquer Method
Another method for computing eigenvalues andeigenvectors of real symmetric tridiagonal matrix is basedon divide-and-conquer
Express symmetric tridiagonal matrix T as
T =
[T1 OO T2
]+ β uuT
Can now compute eigenvalues and eigenvectors of smallermatrices T1 and T2
To relate these back to eigenvalues and eigenvectors oforiginal matrix requires solution of secular equation, whichcan be done reliably and efficiently
Michael T. Heath and Edgar Solomonik Parallel Numerical Algorithms 40 / 42
BasicsPower Iteration
QR IterationKrylov MethodsOther Methods
Jacobi MethodBisection MethodsDivide-and-Conquer
Divide-and-Conquer Method
Applying this approach recursively yieldsdivide-and-conquer algorithm that is naturally parallel
Parallelism in solving secular equations grows asparallelism in processing independent tridiagonal matricesshrinks, and vice versa
Algorithm is complicated to implement and difficultquestions of numerical stability, eigenvector orthogonality,and load balancing must be addressed
Michael T. Heath and Edgar Solomonik Parallel Numerical Algorithms 41 / 42
BasicsPower Iteration
QR IterationKrylov MethodsOther Methods
References
Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, and H. van der Vorst,eds., Templates for the Solution of Algebraic EigenvalueProblems: A Practical Guide, SIAM, 2000
J. W. Demmel, M. T. Heath, and H. A. van der Vorst, Parallelnumerical linear algebra, Acta Numerica 2:111-197, 1993
P. J. Eberlein and H. Park, Efficient implementation of Jacobialgorithms and Jacobi sets on distributed memory architectures,J. Parallel Distrib. Comput., 8:358-366, 1990
G. W. Stewart, A Jacobi-like algorithm for computing the Schurdecomposition of a non-Hermitian matrix, SIAM J. Sci. Stat.Comput. 6:853-864, 1985
G. W. Stewart, A parallel implementation of the QR algorithm,Parallel Comput. 5:187-196, 1987
Michael T. Heath and Edgar Solomonik Parallel Numerical Algorithms 42 / 42
BasicsDefinitionsTransformationsParallel Algorithms
Power IterationPower IterationInverse IterationSimultaneous Iteration
QR IterationOrthogonal IterationQR Iteration
Krylov MethodsKrylov SubspacesArnoldi IterationLanczos IterationKrylov Subspace Methods
Other MethodsJacobi MethodBisection MethodsDivide-and-Conquer