Computing the smallest eigenvalue of symmetric tridiagonal - Lirias

On computing the smallesteigenvalue of

symmetric tridiagonalmatrices

IntroductionQR-iterations on the inverse

An implicit approachConclusions & Future work

Computing the smallest eigenvalueof symmetric tridiagonal matrices

via semiseparable techniques.

Raf Vandebril, Marc Van Barel,Nicola Mastronardi, Steven Delvaux & Yvette Vanberghen

Department of Computer ScienceK.U.Leuven

July 2006

On the orthogonal reductions to semiseparable form





Goal of the talkSemiseparable matricesThe QR-algorithm

Outline

IntroductionGoal of the talkSemiseparable matricesThe QR-algorithm

QR-iterations on the inverse

An implicit approach

Conclusions & Future work







Outline











Accurately computing the smallest eigenvalue

I Suppose we have a symmetric tridiagonal n× n matrix T,with one small eigenvalue.

|λ1| ≈ 10−x with 6 ≤ x ≤ 14

|λi| ≈ 1 with 2 ≤ i ≤ n

The smallest one often suffers from loss in accuracy.I Our aim is to compute the eigenvalue λ1 as accurate as

possible, in linear time.I We will base ourselves on the QR-algorithm.I We will make use of semiseparable matrices. These

matrices are the inverses of tridiagonal matrices.







Outline











A semiseparable matrix

I A matrix A is called semiseparable if all submatrices takenout of the lower triangular part have rank 1. (Similar for theupper triangular part.)

I All the red blocks are of rank 1.× × × × ×× × × × ×× × × × ×× × × × ×× × × × ×

I The inverse of a tridiagonal matrix is semiseparable and

can be computed in O(n) operations.









I All the red blocks are of rank 1.� × × × ×� × × × ×� × × × ×� × × × ×� × × × ×

I The inverse of a tridiagonal matrix is semiseparable andcan be computed in O(n) operations.









I All the red blocks are of rank 1.× × × × ×� � × × ×� � × × ×� � × × ×� � × × ×










I All the red blocks are of rank 1.× × × × ×× × × × ×� � � × ×� � � × ×� � � × ×










I All the red blocks are of rank 1.× × × × ×× × × × ×× × × × ×� � � � ×� � � � ×










I All the red blocks are of rank 1.× × × × ×× × × × ×× × × × ×× × × × ×� � � � �

I The inverse of a tridiagonal matrix is semiseparable and

can be computed in O(n) operations.







InversionSuppose we have a matrix T, inverting this matrix is almoststraightforward using:

I LU-factorization (L is lower bidiagonal, U is upperbidiagonal):

T = LU → S = T−1 = U−1L−1,

with U−1 of upper triangular semiseparable form and L−1

of lower triangular semiseparable form.I QR-factorization (Q is a unitary Hessenberg matrix, R is

upper triangular, with two nonzero superdiagonals):

T = QR → S = T−1 = R−1Q−1,

with Q−1 unitary upper Hessenberg and R−1 an uppertriangular semiseparable matrix of semiseparability rank 2.







Inversion

Or via direct inversion, with

T =

a1 b1b1 a2 b2

b2 a3 b3b3 a4

and

S =

c1v1 c2s1v1 c3s2s1v1 s3s2s1v1

c2s1v1 c2v2 c3s2v2 s3s2v2c3s2s1v1 c3s2v2 c3v3 s3v3s3s2s1v1 s3s2v2 s3v3 v4

.







Outline











The QR-algorithm

I Suppose we have an n× n matrix A. Computing the matrixA, gives us one step of the QR-algorithm with shift κ:

A− κI = QR

A = RQ + κI.

A converges to the largest eigenvalue, and on the bottomright to the smallest eigenvalue, w.r.t. the shift!!

I Repeating this procedure gives us the standardQR-algorithm. This method lets A converge to a diagonalmatrix, with the eigenvalues on the diagonal.

I One step of the QR-method applied on a semiseparable oron a tridiagonal matrix costs O(n) operations.






QR-algorithm with shiftQR-algorithm without shift

Outline

Introduction

QR-iterations on the inverseQR-algorithm with shiftQR-algorithm without shift









Outline

Introduction










QR-algorithm with shift on the inverse

I QR-algorithm with shift on the matrix T and on its inverse S.

I Standard QR-algorithm (with shift) computes first thelargest eigenvalues.

I This results in a lot of flops before the smallest one iscomputed in the tridiagonal case.







Results of QR-algorithm with shiftI The smallest eigenvalue via the semiseparable approach,

is a little bit more accurate (λ1 ≈ 10−12).I But, we do loose accuracy when inverting the tridiagonal

matrix. (Blue is semiseparable, Green is tridiagonal)

1 2 3 4 5 610

−5

10−4

10−3

10−2

10−1

100

Experiments (size*100)

Rel

ativ

e A

ccur

acy







Outline

Introduction










QR-algorithm without shiftI Without shift means no forced convergence to the largest

eigenvalue (λ1 ≈ 10−12).I Both approaches perform equally well.

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 610

−9

10−8

10−7


Rel

ativ

e ac

cura

cy






The explicit approachThe implicit approachCombined QR-iterations without shiftThe implicit QR-algorithm of a tridiagonalAlgoritmic idea

Outline

Introduction


An implicit approachThe explicit approachThe implicit approachCombined QR-iterations without shiftThe implicit QR-algorithm of a tridiagonalAlgoritmic idea to compute the smallest eigenvalue








Outline

Introduction










The QR-algorithm, the explicit approachSuppose we have a shift κ, and we calculate theQR-decomposition of the matrix A− κI:

A− κI = QR

and let us denote with A the matrix:

A = RQ + κI.

The matrix A is the result of performing one step of theQR-algorithm on the matrix A.

When we perform the operations as presented here, this iscalled the explicit approach.







Outline

Introduction










The implicit approach

We can rewrite the formulas from before as:

(A− κI) = QR ⇒ Q−1 (A− κI) = R

substituting R in the next equation

A = RQ + κI

gives usA = Q−1AQ.

When we apply the transformations Q and Q−1 at the sametime, we have an implicit algorithm.







Why an implicit algorithm?

I If only the eigenvalues are desired, we do not need to storeQ.

I We do not lose accuracy due to the adding and subtractingof the shift matrix κI, compared to the explicitQR-algorithm with shift.







Outline

Introduction










Combined QR-iterations

I Inverting causes loss in accuracy!I Suppose we have a tridiagonal matrix T with its inverse

S = T−1.I If S converges to the dominant eigenvalue in S = QHSQ, we

have that:

T = S−1

QHTQ = QHS−1Q =(QHSQ

)−1[T ε

εT λ1

]=

[S ε

εT 1/λ1

]−1







A first attempt

I As seen before, when using the QR-algorithm with shift,we get quick convergence to the largest eigenvalue of S.

I Using now this Q to compute:

T = QHTQ,

would force the matrix T to converge to its smallesteigenvalue.

I Numerical results are not satisfactory.I Due to small rounding errors, causing the factor Q, not to

be exactly equal to the factor Q for T we wanted.







Outline

Introduction










The QR-factorization of tridiagonal matrices

The QR-factorization of the tridiagonal matrix A− κI, consists ofn− 1 Givens transformations performed from the bottom to thetop.

× ×⊗ × ×

× × ×× × ×

× ×

I Apply a Givens transformation

on the left.I Involving row 1 and 2.








The QR-factorization of the tridiagonal matrix A− κI, consists ofn− 1 Givens transformations

× × ×× ×⊗ × ×

× × ×× ×











× × ×× × ×

× ×⊗ × ×

× ×











× × ×× × ×

× × ×× ×⊗ ×











× × ×× × ×

× × ×× ×

×

I We have now an uppertriangular matrix R.







Implicit QR-algorithm for tridiagonal matrices

As mentioned before we want to apply the transformations atboth sides at the same time

× ×× × ×

× × ×× × ×

× ×

I Apply the first Givens

transformation on both sidesof the matrix.

I Involving row 1 and 2.I Involving column 1 and 2.









× × ⊗× × ×⊗ × × ×

× × ×× ×


to annihilate the bulge.I Involving row 2 and 3.I Involving column 2 and 3.









× ×× × × ⊗

× × ×⊗ × × ×

× ×











× ×× × ×

× × × ⊗× × ×⊗ × ×











× ×× × ×

× × ×× × ×

× ×

I We have again a tridiagonalmatrix.







The implicit Q-theorem

I In the implicit approach, it seemed that only one Givenstransformation was needed to start the process.

I So the algorithm is as follows:I Compute the first Givens transformation of the

QR-factorization of T − κI.I Apply it on both sides of the matrix T.I Chase the bulge.

I The implicit Q-theorem proves that this approach performsa step of the QR-algorithm on T.







Outline

Introduction










The idea

I We know that if S converges to the largest via theQR-algorithm, that T converges to the smallest.

I Determine the first Givens transformation of the implicitapproach for T, via the QR-factorization of S.







Algorithm 1

A first algorithm has the following structure:1. Invert the matrix T. Denote S = T−1.2. Compute the initial Givens transformation for S.3. Perform the implicit QR-step on T.

Store this in T.4. Perform the implicit QR-step on S.

Store this in S.5. Start again at step 2.







Remarks on algorithm

Remarks on this algorithm:1. Due to small rounding errors, we get more and more

divergence between the intial Givens of S and the initialGivens we need in T.

2. This can result in slow convergence.3. Slow convergence causes loss in accuracy.







Algorithm 2

A more stable approach1. Invert the matrix T. Denote S = T−1.2. Compute the initial Givens transformation for S.3. Perform the implicit QR-step on T.

Store this in T.4. Start again at step 1.

The complexity is similar to the complexity of the first algorithm.







Remarks on algorithm

Remarks on this algorithm:1. More stable due to the forced inversion of T.2. This leads to a more accurate first Givens transformation.3. Faster convergence.4. Less loss in accuracy.







Remarks on algorithmBlue is the new approach, green represents the traditionalapproach.

1 2 3 4 510

−5

10−4

10−3

10−2

10−1

100


Rel

ativ

e A

ccur

acy






Outline

Introduction









Conclusions

I No dramatic increase in accuracy.I Improving convergence speed can be done via

I More accurate inversion of the tridiagonal.I Using shifted QR for the semiseparable to determine the

first Givens.I Improving accuracy can be done via

I Iterative refinement.I More accurate first Givens transformation in the implicit

approach, i.e. more accurate inversion.


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