IDR a brief introduction Martin H. Gutknecht ETH Zurich, Seminar - - PowerPoint PPT Presentation

IDR — a brief introduction

Martin H. Gutknecht

ETH Zurich, Seminar for Applied Mathematics Minisymposium “Induced Dimension Reduction (IDR) methods” SIAM Conf. on Applied Linear Algebra, Monterey, CA, USA, Oct. 27, 2009

Prerequisites History IDR basics Case s=1 Case s>1 Conclusions

Outline

Prerequisites History IDR basics Case s=1 Case s>1 Conclusions

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Prerequisites History IDR basics Case s=1 Case s>1 Conclusions

Prerequisites: Krylov (sub)space solvers

Given: linear system Ax = b, initial approx. x0 ∈ CN. Construct: approximate solutions (“iterates”) xn and corresponding residuals rn :≡ b − Axn with xn ∈ x0 + Kn(A, r0) , rn ∈ r0 + AKn(A, r0) where r0 :≡ b − Ax0 is the initial residual, and Kn :≡ Kn(A, r0) :≡ span {r0, Ar0, . . . , An−1r0} is the nth Krylov subspace generated by A from r0. We can, e.g., construct xn such that rn is minimal.

• Conjugate Residual (CR) method (Stiefel, 1955),
• GCR and GMRES.

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Some further Krylov space solvers are based on other

• rthogonal or oblique projections:

Conjugate Gradient (CG) method (Hestenes/Stiefel, 1952): rn ∈ r0 + AKn , rn ⊥ Kn . Biconjugate Gradient (BICG) method (Lanczos, 1952; Fletcher, 1976): rn ∈ r0 + AKn , rn ⊥ Kn :≡ Kn(A

⋆,

r0) . ML(s)BICG method (M.-C. Yeung and T. F. Chan, 1999): rsj ∈ r0 + AKsj , rsj ⊥ Kj(A

⋆,

R0) :≡

s

• i=1

Kj(A

⋆,

r(i)

0 ) .

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Prerequisites: residual polynomials

rn ∈ r0 + AKn(A, r0) implies that ∃ρn ∈ Pn , ρn(0) = 1 : rn = ρn(A)r0 . Means roughly: rn is small if |ρn(t)| is small at the eigenvalues of A. Means also: “everything” can be formulated in terms of residual polynomials.

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Special cases: In CG the residual polynomials are orthogonal polynomials (OPs) w.r.t. a weight function determined by EVals of A (symmetric) and by r0. In BICG the residual polynomials are formal orthogonal polynomials (FOPs). Lanczos polynomials. In (Bi)Conjugate Gradient Squared (CGS), ρCGS

n

=

• ρ

BICG

n

2 . in BICGSTAB, ρ

BiCGSTAB

n

= ρ

BICG

n

Ωn , where Ωn(t) :≡ (1 − ω1t) · · · (1 − ωnt). Here, at step n, ωn is chosen to minimize the residual on a straight line.

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History of IDR: references

• P. WESSELING AND P. SONNEVELD, Numerical

experiments with a multiple grid and a preconditioned Lanczos type method, in Approximation Methods for Navier-Stokes Problems, R. Rautmann, ed., Lecture Notes in Mathematics, vol. 771, Springer, 1980, pp. 543–562. Introduce Induced Dimension Reduction (IDR) method, attributed to Sonneveld (“Public. in preparation”), on 3 1

2

pages: a “Lanczos-type method” for nonsymmetric linear systems which does not require AT.

• P. SONNEVELD, CGS, a fast Lanczos-type solver for

nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 10 (1989), pp. 36–52. Received Apr. 24, 1984. Introduces (Bi)CG Squared.

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• H. A. VAN DER VORST, Bi-CGSTAB: a fast and smoothly

converging variant of Bi-CG for the solution of nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 13 (1992), pp. 631–644. Received May 21, 1990. Presented at Householder

• Tylosand. 1st preprint: “CGSTAB: A more smoothly

converging variant of CG-S”, coauthored by P . Sonneveld. M.-C. YEUNG AND T. F. CHAN, ML(k)BiCGSTAB: a BiCGSTAB variant based on multiple Lanczos starting vectors, SIAM J. Sci. Comput., 21 (1999), pp. 1263–1290. Received May 16, 1997. Introduce first ML(k)BiCG and then ML(k)BiCGSTAB — versions of BICG and BICGSTAB, resp., with “multiple left (shadow) residuals”. Fundamental idea. Astonishing numerical results. Some details complicated and not so well explained.

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• P. SONNEVELD AND M. B. VAN GIJZEN, IDR(s): a family of

simple and fast algorithms for solving large nonsymmetric systems of linear equations, Report 07-07, Department of Applied Mathematical Analysis, Delft University of Technology; SIAM J. Sci. Comput., 31 (2008), pp. 1035–1062. Generalizing IDR ≈ IDR(1) to IDR(s). A fundamental

• generalization. Very good numerical results.

Detailed description of method, connection to BICGSTAB.

• G. SLEIJPEN, P. SONNEVELD, AND M. B. VAN GIJZEN,

Bi-CGSTAB as an induced dimension reduction method, Report 08-07, Department of Applied Mathematical Analysis, Delft University of Technology. Partly new view; explores connection to BiCGSTAB and ML(k)BiCGSTAB from a partly different point of view.

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• M. B. VAN GIJZEN AND P. SONNEVELD, An elegant IDR(s)

variant that efficiently exploits bi-orthogonality properties, Report 08-21, Department of Applied Mathematical Analysis, Delft University of Technology. Uses the freedom in the choice of the “intermediate” residuals to come up with a new version of IDR(s) that is slightly more efficient and particularly ingenious.

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This talk is based on:

• M. H. GUTKNECHT, IDR explained. To appear in ETNA.

Further papers by: KUNIYOSHI ABE AND GERARD SLEIJPEN, TIJMEN COLLIGNON, YUSUKE ONOUNE AND SEIJI FUJINO, VALERIA SIMONCINI AND DANIEL SZYLD, MASAAKI TANIO AND MASAAKI SUGIHARA MS38, Wes 2:45pm, MAN-CHUNG YEUNG, MHG AND JENS-PETER ZEMKE, ... plus many more on applications.

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Prerequisites History IDR basics Case s=1 Case s>1 Conclusions

IDR(s) basics: the setting

Given: linear system Ax = b ∈ CN, initial approx. x0. Let: r0 :≡ b − Ax0 , Km :≡ Km(A, r0) :≡ span {r0, Ar0, . . . , Am−1r0} , ν such that G0 :≡ Kν invariant , S ⊂ CN linear subspace of dimension N − s , for j = 1, 2, . . . : choose ωj = 0 and let Gj :≡ (I − ωjA)(Gj−1 ∩ S) , for n = nj, . . . , nj+1 − 1 : choose xn such that rn ∈ Gj ∩ (r0 + AKn

• ⊂ Kn+1

) . Note: Typically nj+1 := nj + s + 1 .

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IDR(s) basics: the spaces Gj (case s = 1)

G0 = R3 G0 ∩ S = S I − ω1A G1 G1 ∩ S G2 ∩ S = {0} G2 I − ω2A

Gj :≡ (I − ωjA)(Gj−1 ∩ S)

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IDR(s) basics: IDR theorem

Recall: Gj :≡ (I − ωjA)(Gj−1 ∩ S) , rn ∈ Gj ∩ (r0 + AKn) . Genericness assumption: S ∩ G0 contains no eigenvector of A .

THEOREM (IDR THEOREM (WES/SON80,SON/VGI07))

Gj Gj−1 unless Gj−1 = {o} . Consequently: Gj = {o} for some j ≤ N. IDR Thm. suggests: rn = o once rn ∈ GN , i.e., j = N . But typically rn ∈ Gj when n = j(s + 1) . However, normally rn = o once n = N . Hence: IDR Thm. strongly underestimates convergence rate.

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IDR(s) basics: what’s different?

Most currently used KSS (= Krylov subspace solvers) are based on a different kind of “induced dimension reduction”: rn ∈ L⊥

n ∩ (r0 + AKn(A, r0)) ,

where, e.g., Ln = Kn(A, r0) (CG) , Ln = A Kn(A, r0) (CR, GCR, GMRES) , Ln = Kn :≡ Kn(A

⋆,

r0) (BiCG) . IDR: Gj is not an orthogonal complement of a Krylov subspace. However, due to form of the recursion for {Gj}, Gj turns out to be the image of an orthogonal complement of a Krylov subspace.

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IDR(s) basics: recursions for {rn}

Recall: Gj :≡ (I − ωjA)(Gj−1 ∩ S) . Wanted: rn+1 ∈ Gj ∩ (r0 + AKn+1) . = ⇒ rn+1 := (I − ωjA) vn , vn ∈ Gj−1 ∩ S ∩ (r0 + AKn) , = ⇒ vn := rn −

ι(n)

• i=1

γ(n)

i

∆rn−i = rn − ∆Rn cn , (1) where s ≤ι(n) ≤ n − nj−1 , ( ∆rn−i ∈ Gj−1) ∆rn :≡ rn+1 − rn , ∆Rn :≡ ∆rn−1 . . . ∆rn−ι(n)

• ,

cn :≡

• γ(n)

1

. . . γ(n)

ι(n)

T .

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Recall (1): vn := rn −

ι(n)

• i=1

γ(n)

i

∆rn−i = rn − ∆Rn cn ∈ Gj−1 ∩ S . Since dim S = N − s, there is P ∈ CN×s s.t. S⊥ = R(P) : vn ∈ S ⇐ ⇒ vn ⊥ S⊥ = R(P) ⇐ ⇒ P⋆vn = o . To achieve this, the term ∆Rn cn in (1) must be the oblique projection of rn into R(∆Rn) along S. In order that this projection is uniquely defined, we need P⋆ ∆Rn to be nonsingular. Then vn := rn − ∆Rn (P⋆ ∆Rn)−1 P⋆rn

• ≡:cn

= rn − ∆Rn cn . (2) We need ι(n) = s to make P⋆ ∆Rn a square matrix.

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IDR(s) basics: the first two steps (case s = 1)

G0 = R3 r2 = (I − ω1A)v1 r0 r1 v1 G0 ∩ S = S I − ω1A v2 r3 G1 G

1

∩ S

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IDR(s) basics: all the residuals (case s = 1)

G0 = R3 r2 = (I − ω1A)v1 r0 r1 v1 G0 ∩ S = S I − ω1A v2 r3 G1 v3 G

1

∩ S G2 ∩ S = {0} = {v5} G2 I − ω2A v4 r4 r5

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IDR(s) basics: choice of ωj

Recall: rn+1 = (I − ωjA) vn (3) Here, ωj is fixed for n + 1 = nj, . . . , nj+1 − 1. So, only for n + 1 = nj, we may choose ωj s.t. rn+1 is minimal among all r of the form r = (I − ωjA) vn , i.e., r ⊥ Avn : ωj :≡ Avn, vn Avn2 .

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IDR(s) basics: recursions for {xn}

Note 1: vn ∈ r0 + AKn = ⇒ ∃ x′

n

s.t. vn = b − Ax′

n ,

i.e., vn is the residual of an “intermediate” iterate x′

n ∈ x0 + Kn .

Note 2: ∆rn = −A ∆xn, ∆Rn = −A ∆Xn, Hence: vn := rn − ∆Rn cn = ⇒ x′

n := xn − ∆Xn cn ,

(4) rn+1 := (I − ωjA) vn = ⇒ xn+1 := ωjvn + x′

n .

(5) There are several ways to rearrange these four recursions and to combine them with the iterate-residual relationships; see [Sle/Son/vGi08].

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The two formulas rn+1 := (I − ωjA) vn , vn := rn − ∆Rn cn can be combined into rn+1 := rn − ∆Rn cn − ωjA vn

• =∆rn

. (6) Along with it: xn+1 := xn − ∆Xn cn + ωjvn

• =∆xn

. (7) We may also combine the second formula and ∆rn = −A ∆xn. This is the choice in the “prototype algorithm” of [Son/vGi07]. So, there are many ways to implement IDR(s) — and more to come!

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Prerequisites History IDR basics Case s=1 Case s>1 Conclusions

IDR(s) basics: characterization by orthogonality

THEOREM (SON/VGI07, SLE/SON/VGI08)

Let Ω0(t) :≡ 1 , Ωj(t) :≡ (1 − ω1t) · · · (1 − ωjt) ∈ P◦

j , where

P◦

j :≡ {polyns. of degree ≤ j that are 1 at 0}. Then

Gj =

• Ωj(A)w
• w ⊥ Kj(A

⋆, P)

• ≡:

Kj

• = Ωj(A)
• Kj(A

⋆, P)

⊥

• =

K⊥

j

. Note: Kj is the left-hand side (LHS) block Krylov space that appears in the block Lanczos process with LHS block size s. Note: We may have Lanczos breakdowns and a collapsing block Krylov space (which requires deflation).

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Since rn ∈ Gj ∩ (r0 + AKn) , and since the residual polynomials must have full degree, we have for n = nj, . . . , nj+1 − 1 : rn = Ωj(A) wn , wn ∈ (r0 + AKn−j) ∩ K⊥

j ,

wn ∈ Kn−j . (8) Generically, for n = nj = j (s + 1) , where wn ∈ (r0 + AKjs) and wn ⊥ Kj with dim Kj = js , there is a unique wn satisf. (8). But the s other vectors wn with nj ≤ n < nj+1 are not uniquely determined by (8).

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s = 1: IDR(1) ∼ BICGSTAB

Every other set of vectors (wn, rn, vn−1, xn, . . . ) is uniquely determined — up to the choice of the parameters ωj. Normally, the latter are chosen as in BICGSTAB, and thus r2j = r

BiCGSTAB

j

, x2j = x

BiCGSTAB

j

, w2j = r

BICG

j

, where r

BICG

j

is the jth residual of BICG: r

BICG

j

= ρj(A)r0 . Recursions (4) and (5), with γn :≡ γ(n)

1

= P, rn / P, ∆rn−1 : vn := (1 − γn)rn + γnrn−1 , x′

n := (1 − γn)xn + γnxn−1 ,

rn+1 := (I − ωjA) vn , xn+1 := x′

n + ωjvn .

(9)

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s = 1: polynomial recursions

rn = Ωj(A)wn =

• Ωj(A)ρj(A)r0

if n = 2j , Ωj(A) ρj+1(A)r0 if n = 2j + 1 , vn = Ωj−1(A)wn+1 =

• Ωj−1(A)ρj(A)r0

if n = 2j − 1 , Ωj−1(A) ρj+1(A)r0 if n = 2j , Inserting these formulas into vn = (1 − γn)rn + γnrn−1 we get, after a short calculation, for n = 2j and n = 2j + 1, respectively,

• ρj+1(t) := (1 − γ2j) (1 − ωjt) ρj(t) + γ2j

ρj(t) , ρj+1(t) := (1 − γ2j+1) ρj+1(t) + γ2j+1 ρj(t) . (10) Recall: w2j = ρj(A)r0 ⊥ Kj and w2j+1 = ρj(A)r0 ⊥ Kj .

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Prerequisites History IDR basics Case s=1 Case s>1 Conclusions

BICG uses in its standard version coupled two-term recursions: r

BICG

j+1 := r

BICG

j

− αjAv

BICG

j

, v

BICG

j+1 := r

BICG

j+1 + βjv

BICG

j

. The corresp. recursions for ρj and σj are ρj+1(t)

⊥Pj

:= ρj(t)

• ⊥Pj−1

−αj tσj(t)

⊥Pj−1

, σj+1(t)

⊥t Pj

:= ρj+1(t)

⊥Pj

+βj σj(t)

• ⊥t Pj−1

. In contrast, in IDR(1), by (10),

• ρj+1(t)

⊥Pj−1

:= (1 − γ2j) (1 − ωjt) ρj(t)

• ⊥Pj−2

+γ2j ρj(t)

• ⊥Pj−2

, ρj+1(t)

⊥Pj

:= (1 − γ2j+1) ρj+1(t)

⊥Pj−1

+γ2j+1 ρj(t)

• ⊥Pj−1

.

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Prerequisites History IDR basics Case s=1 Case s>1 Conclusions

Comparing the recursions for (ρj, σj) with those for (ρj, ρj) we easily see: (1 − γ2j+1)

• ρj+1(t) − ρj(t)
• = −αj t σj(t) ,
• r,
• ρj+1(t) = ρj(t) −

αj 1 − γ2j+1 t σj(t) ,

• r,

r2j+1 = r2j − αj (1 − γ2j+1) A Ωj(A)v

BICG

j

• ≡: s

BiCGSTAB

j

. This formula expresses the odd indexed IDR(1) residuals in terms of quantities from BICGSTAB and the coefficient γ2j+1.

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Prerequisites History IDR basics Case s=1 Case s>1 Conclusions

• ρj+1

ρj tσj −αj t σj ρj+1 −α′

j t σj

• ρj+1(t) = ρj(t) −

≡: α′

j

• αj

1 − γ2j+1 t σj(t) , BICG/BICGSTAB: ρj+1(t) := ρj(t) − αjtσj(t) , IDR(1): ρj+1(t) := (1 − γ2j+1) ρj+1(t) + γ2j+1 ρj(t) .

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s = 1: Comments and conclusions

• IDR(1) can be viewed as a minor variation of BICGSTAB.
• It is not clear, why one or the other should be more stable.
• In fact, the existence of all even indexed IDR(1) residuals

requires (like the BIORES version of BICG) that no Lanczos breakdowns and no pivot breakdowns occur.

• The smoothing step breakdown (ωj = 0) is also the same

and can be treated easily by choosing a non-optimal ωj.

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Prerequisites History IDR basics Case s=1 Case s>1 Conclusions

s > 1: IDR(s) ∼ ML(s)BICGSTAB

• Relation

IDR(1) ∼ BICGSTAB ⇐ = BICG is matched by relation IDR(s) ∼ ML(s)BICGSTAB ⇐ = ML(s)BICG.

• If the parameters ωj were chosen the same in IDR(s) and

ML(s)BICGSTAB every (s + 1)th iterate were the same in both methods — but normally the parameters are not chosen the same.

• ML(s)BICG and ML(s)BICGSTAB are due to Man-Chung

Yeung and Tony Chan ’97/’99SISC. Connection to nonsym. block Lanczos [Aliaga/Boley/Freund/Hernández ’96/’99MC].

• ML(s)BICGSTAB requires a few more inner products and

vector updates than IDR(s) and may be less stable.

• IDR(s) is algorithmically simpler and more flexible.

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Prerequisites History IDR basics Case s=1 Case s>1 Conclusions

• The fundamental discovery due to Yeung and Chan is that,

in the framework of Lanczos-type product methods, multiple left projections can both speed up the convergence and reduce the MV count (per “degree” of wn, which is what is determined by orthogonality). Sonneveld and van Gijzen rediscovered this 10 years later.

• Yeung/Chan paper is ok, but the derivation of the

recursions is complicated and “unusual” due to use of a non-biorthogonal basis for Kj and complex manipulations for reducing cost. Amazingly, their Matlab program

is only 187 lines (incl. 30 lines of comments).

• The new paper of Yeung explains the method better and

improves it.

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Prerequisites History IDR basics Case s=1 Case s>1 Conclusions

Conclusions

IDR(1) is as good as BICGSTAB. IDR(s) is as good as ML(s)BICGSTAB. The IDR(s) recurrence coefficients are simpler to compute than those of ML(s)BICGSTAB. Typically, IDR(s) and ML(s)BICGSTAB outperform other methods for a nonsymmetric problem. The new IDR(s)BiO of van Gijzen and Sonneveld performs even slightly better. IDR-like generalizations of BiCGStab2 and BiCGStab(ℓ) are even more promising, in particular if A is real, but has (strongly) non-real eigenvalues.

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Prerequisites History IDR basics Case s=1 Case s>1 Conclusions

• M. H. GUTKNECHT, IDR explained, Research Report 2009-14,

Seminar for Applied Mathematics, ETH Zurich, March 2009.

• G. L. G. SLEIJPEN, P. SONNEVELD, AND M. B. VAN GIJZEN,

Bi-CGSTAB as an induced dimension reduction method, Report 08-07, Department of Applied Mathematical Analysis, Delft University of Technology, 2008.

• P. SONNEVELD, CGS, a fast Lanczos-type solver for

nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 10 (1989), pp. 36–52. Received Apr. 24, 1984.

• P. SONNEVELD AND M. B. VAN GIJZEN, IDR(s): a family of

simple and fast algorithms for solving large nonsymmetric systems of linear equations, Report 07-07, Department of Applied Mathematical Analysis, Delft University of Technology, 2007.

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Prerequisites History IDR basics Case s=1 Case s>1 Conclusions

• P. SONNEVELD AND M. B. VAN GIJZEN, IDR(s): A family of

simple and fast algorithms for solving large nonsymmetric systems of linear equations, SIAM Journal on Scientific Computing, 31 (2008), pp. 1035–1062. Received Mar. 20, 2007.

• H. A. VAN DER VORST, Bi-CGSTAB: a fast and smoothly

converging variant of Bi-CG for the solution of nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 13 (1992),

• pp. 631–644.

Received May 21, 1990.

• P. WESSELING AND P. SONNEVELD, Numerical experiments with

a multiple grid and a preconditioned Lanczos type method, in Approximation methods for Navier-Stokes problems (Proc. Sympos., Univ. Paderborn, Paderborn, 1979), vol. 771 of Lecture Notes in Math., Springer, Berlin, 1980, pp. 543–562.

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M.-C. YEUNG AND T. F. CHAN, ML(k)BiCGSTAB: a BiCGSTAB variant based on multiple Lanczos starting vectors, SIAM J. Sci. Comput., 21 (1999), pp. 1263–1290. Received May 16, 1997, electr. publ. Dec. 15, 1999.

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