BLAS Specification Revisited

Linda KaufmanWilliam Paterson University

Wish list

• Simultaneous Orthogonal Transformations– Generation– Application

• Simultaneous elementary transformations• Simultaneous gemv with different matrices• Simultaneous Householders• Symmetric rank k update that manufacturers

might want not balk at implementing

Simultaneous Orthogonal transformations

A. QZhes- reduction of A to Hessenberg form and B to triangular formB. QR iteration for symmetric tridiagonal eigenvaluesC. Reduction of narrow banded matrix to tridiagonal form in order to solve an

eigenvalue problem Ax=λxA. Approximation of 1 dimensional pde with a Rayleigh Ritz Galerkin

approach using cubic or quintic B splinesB. Periodic boundary conditions of 1 dimensional pde with finite elementC. Coupling several problems with 1 dimensional pde as in designing optical

fibersD. Ax = λB x, A and B symmetric, B positive definite, A and B banded

A. B tridiagonal mass matrix in finite element approximation of 1 D problem,

E. Banded singular value decomposition-S. Rajamanickam thesis under Tim Davis

F. To prevent fill in when pivoting in Symmetric indefinite banded factorizationA. Optimization problem with negative curvature

QZ phase 1 reduction of A to upper Hessenberg B to triangular for solving nonsymmetric Ax = λB x

Assume we have used orthogonal transformations to reduce B to triangular and have applied them to A. We now have

A AB B A B

In LAPACK get rid of elements of A in the order of But there are independent operations that can

be done simultaneously using the ordering

In general 2n simultaneous operations. B. Kagstrom & Dackland, 1999One can look at these as blocks or individual elements

→ →

IMTQL1- finding the eigenvalues of Ax=λx for A symmetric tridiagonal

1 Compute shift μ, form B =A- μI2. Find Q1 that annihilates B21 and form Q1B Q1

T

3. Chase unwanted element down matrix

Parallel QR- keep on determining shift and do simultaneous chases. Van-de Gijn(1993)- 3 times as many chases but Kaufman showed can get factor of 2 reduction (1994)

→→

→

Diagram of annihilation using Given’s rotations for banded eigenvalue

Eventually every kth row have element that could be annihilated for r 2k-1 diagonals (Kaufman-1984) implemented in Lapack-(Christian Bischof, Bruno Lang, and XiaobaiSun -SBR toolbox 2000)

Saw reduction by factor of 5 for narrow bands on Cray

Would like to have been able to generate Givens rotations simultaneously- killed by manufacturers

Diagrams of parallel Crawford for reducing symmetric tridiagonal Ax = λB x, to standard eigenvalue problem

AA

B B

→

→

→

Simultaneous Stabilized elementary transformations• Simultaneously factoring banded linear systems

• Tinvit- Given several eigenvalues (λ1, λ2, … λk) of a tridiagonal system determine their eigenvectors. Solve A- λ1

I, A- λ2 I, … A- λk I simultaneously

• Shot gun bisection

• Two dimensional separable elliptic PDE’s solved using marching(Bank), Rayleigh Ritz Galerkin(Kaufman and Warner), or Collocation(Fairweather).• Matrix has form A = (Tensor Product where

S’s and M’s are banded. )

• Queueing Problems leading to separable matrices(Kaufman, 1983)

• Symmetric Indefinite using stabilized elementary to prevent fill-in

Separable matrix in solving 2 dimensionalMatrix hasthe form A = (Tensor Product ) Where Sx and Mx are m x m banded and Sy and My are n xn banded, Mx , My, and Sy are symmetric and My is positive definite

Need to Solve Av = f where A has mn rows and columns Algorithm:(1) Find D and Z such that ZT My Z = I and ZT Sy Z = D Generalize eigenvalue problem(2) Compute g = f

(3) Solve the n banded systems given by h = g

(4)Compute v = h

Steps 2 and 4 just use Matrix-Matrix multiply and are fast. Sometimes Z and D are known apriori like for Poisson’s

equation with uniform grid.One can reduce Step(3) by factor of 4 by using simultaneous

axpy’s.

Queueing Problems

Often working with singular matrices A with the form

Where the B’s are not symmetric but might have symmetric zero structure, there exists matrices Q and Z such QBjZ is diagonal. Usually all the B’s are the identity matrix except for one. The variable q denotes the number of queues. The problems get large quickly. If one has 10 waiting spaces in q queues the number of variables is 0(10q)

As in the pde case, one can reduce the problem using generalized eigendecompositions to diagonal blocks containing tridiagonal matrices, which here could be unsymmetric-

Symmetric rank k updatesOriginally updates could have the form A= A + α XXT

Could not accommodate the quasi- Newton BFGS update of the approximate Hessian in optimization

LAPACK did not use it and instead treated the lower triangular part as shown where the rectangles used GEMM and small lines that used DGEMV

Could use simultaneous DGEMV here

Workarounds for symmetric updates

At 2011 Householder conference Jennifer Scott suggested adding extra space so that one can use GEMM’s throughout

For symmetric indefinite linear systems, the reduction uses either A=A+ YDYT where D is either 1 x 1 or 2 x 2.

For block D would be a sequence of 1 x 1 or 2 x 2.

2002 Blas suggest A=A + YJYT where J was tridiagonal- I don’t know of any implementations.

Perhaps better to have A=A+XYT but only update triangular part of A

Symmetric banded factorization

For symmetric banded matrices, Kaufman’s retraction algorithm requires (2m+1)n space even though the original matrix can be specified using (m+1)n.

The extra space is to store complications for 2 x 2.

Thus one can imagine the image below and the stuff above diagonal is just scratch space with 1 x 1’s.

Bunch Kaufman for symmetric indefinite non banded

Partition A as

Bandwidth spread with Bunch-Kaufman on banded matrix because of pivoting for stability

Where D is either 1 x 1 or 2 x 2Reset B to B’ = B – Y D-1 YT

when deleting Y’s

Choice of dimension of D depends on magnitudes of a11 versus other elements

Continue with B’ and partitionit as above

Banded algorithm based on B-K

1) Let c = |ar 1 | = max in abs. in col. 12) If |a11 | >= w c, use a 1 x 1 pivot. Here w is a scalar to balance

element growth, like 1/3Else

3)Let f= max element in abs. in column r 4) If w c*c <= |a11 | f, use a 1 x 1 pivot

Else5)interchange the rth and second rows and columns of A6) Do a sequence of orthogonal or elementary

transformation to prevent fill-in while performing a 2 by 2 pivot7) Perform a 2 x 2 pivot

Never pivot with 1 x 1

Pivoting for stability can ruin bandwidth

Worst case r =m, what happens in pivotingx x x x x xx x x x x x x a b c dx x x x x x x xx x x x x x x x xx x x x x x x x x xx x x x x x x x x x x x x x x x x x a x x x x x x x x x b x x x x x x x x c x x x x x x x d x x x x x x x x x x x x

Reset B’ = B – Y D-1 YT= YZLet Z = D-1 YT x x x x x x x x x

p q r s x x x x x .

Then B’ looks like

x x x x x x bp cp dp which by the x x x x x x 0 0 0x x x x x x x cq dq x x x x x x x cq dqx x x x x x x x dr elimination of 1 x x x x x x x x drx x x x x as bs cs ds x x x x x x bt ct dtx x x x x x x x x x element x x x x x x x x x xx x x as x x x x x x x x x x x x x x x xbp x x bs x x x x x x becomes 0 x x bt x x x x x xcp cq x cs x x x x x x 0 cq x ct x x x x x x dp dq dr ds x x x x x x 0 dq dr dt x x x x x x x x x x x x x x x x x x

Continuing in this way gets us back to band form

Partition A as

In practice- Pretreat Z to make zeroes so that rank 2 change does not produce zeroes outside the band

Partition A as where D is 2 X 2

Let Z = D-1 YT

Reset B’ = QT(B – Q Z)Q= QTB Q-HG, Q from fixupWhere H=QTY and G =Z Q

Construct Q so that G= Z Q looks like

Use a sequence of Givens transformations or stabilized planar elementary transformations to form Q so banded structure of QTBQ is not upset.

Because H= QY has form

HG will not extendbeyond band

x x x x x x

x x x0 0 0

x x

x xx x

xx

x

00

0

Comparison with Lapack on positive definiten=2000 posdef

mine-no block dgbtrf dgbtf2 dpbtrf dpbtf2 mine,nb=16

100 0.223 0.266 0.389 0.17 0.218 0.13

200 0.782 0.873 1.46 0.67 0.773 0.382

300 1.64 1.65 3.12 1.49 1.62 0.834

400 2.78 2.41 5.28 2.59 2.76 1.25

500 4.2 3.78 8.14 3.9 4.13 1.76

600 5.61 4.81 11.31 5.62 5.63 2.343

700 7.35 6.3 15.36 7.88 7.33 2.977

0 100 200 300 400 500 600 700 8000

2

4

6

8

10

12

14

16

18

mine-dsyrdgbtrfdgbtf2dpbtrfdpbtf2mine,nb=16

Block version on random matrices-n=2000

mnonblock-retraction

block retraction dgbtf2 dgbtrf 2x2 maxr ave

100 0.327 0.315 0.682 0.451 444 49200 0.986 0.81 2.58 1.3 315 98300 2.08 1.79 5.22 2.58 365 141400 3.37 2.6 8.92 3.93 327 201500 5.45 4.38 13.97 5.85 370 231600 7.19 5.61 24.84 7.64 344 299700 10.23 8.46 37.03 9.85 421 293

Only blocking for 1 x 1, stop accumulating when a 2 x 2 is reached. Elementary transformations for “pretreating” Z. with 2 x 2

0 50000 100000 150000 200000 2500000

1

2

3

4

5

6

retractiondgbtrf

Time as a function of number of planar transformations m=400, n=2000.

Possible ways to speedup retraction for consecutive 2 x 2s:

Each column involves 2 full daxpys plus orthogonal transformations or cut up daxpys to the same column

(1) marching-(1) work on column i+j when elimination starts at i(2) Work on column i+j-1 with elimination starting at i+2(3) Work on column i+j-2 with elimination starting at i+4Requires simultaneous dgemvs or daxpys

(2) 2 sets of transformations(1) Cut up daxpy or orthogonal transformation applied to i+j

stemming from i(2) Dgemv involving 4 columns (i,i+1,i+2,i+3) applied to i+j(3) Cut up daxpy applied to i+j stemming from i+2

Back to requesting simultaneous orthogonal or elementary transformations

Van de Geign to the rescue