potpourri on (fast algorithms for) structured matricesolshevsky/talks/potpourri_02.pdfthe themes...

Potpourri on(Fast Algorithms for)Structured Matrices

Vadim Olshevsky

GSU

Potpourri on(Fast Algorithms for)Structured Matrices – p.1/80

Friends and Colleagues

Parts of this work was done jointly with

A.Shokrollahi (Bell Labs and DigitalFountain), VictorPan (CUNY), and Michael Stewart (ANU and nowGSU).

Other relevant work was done jointly with:

I.Gohberg (Tel Aviv), T.Kailath (Stanford);

Georg Heinig (Germany), Leonid Lerer (Technion),Dario Fasino (Italy.


The themes

Introduction and Some Examples.

Forward stable Cauchy solver.

Backward stable Hankel solver.

Displacement structure .

Tangential interpolation of the Nevanlinna-Pick type .

List decoding of the R-S codes .


Some examplesToeplitz,

� � ��

Hankel,

� ��

Vandermonde,

� ��

Polyn, Vand. � � � � � � ��

Cauchy,

� � � ��

Pick,

� � � � � � � ��

Also Bezoutians, Sylvester matrices, controllability,observability, Krylov matrices and others.Also their inverses, products, Schur complements.



� � ��

Hankel,

� � ��

��

� � � � � � � !� �

� � � � � � � ...... . . . . . . . . . ...... . . . � � � � �� !� � � � � �"

#�#�#�#�#�#�#�#$Vandermonde,

� �� Polyn, Vand.

� � � � � � ��

Cauchy,

� � � �� Pick,

� � � � � � � ��




� � ��

Hankel,

� � ��

��%�

� � & !� � � & . . . ! & . . . . . . ......

!� � ! & !� ' !� � ! & !� ' & !� &

Vandermonde,

� �� Polyn, Vand.

� � � � � � ��

Cauchy,

� � � �� Pick,

� � � � � � � ��




� � ��

Hankel,

� ��

Vandermonde,

� ��

Polyn, Vand. � � � � � � ��

��

( � � � &� � !� ��

......

......

......

......( � ! � & ! � !� �!

"#�#�#�#�#$

Cauchy,

� � � �� Pick,

� � � � � � � ��




� � ��

Hankel,

� � ��

Vandermonde,

� ��

Polyn, Vand. � � � � � � ��

��%�

� � �� !� � ��

......

......

......� � �� ! � � � �� ! � � !� � �� ! �

Cauchy,

� � � �� Pick,

� � � � � ��




� � ��

Hankel,

� ��

Vandermonde,

� ��

Polyn, Vand. � � � � � � ��

Cauchy,

� � � ��

Pick,

� � � � � � � ��

��

��)� � ) ��)� �+*

......��* � � ) ��* � �*

"#�#$




� � ��

Hankel,

� ��

Vandermonde,

� ��

Polyn, Vand. � � � � � � ��

Cauchy,

� � � ��

Pick,

� � � � � � � ��

��

� � �), �� )� )� � � ) � � �), ��*� )� � �*

......� � �* , �� )�+* � � � ) � � �* , ��*� * � � �*

"#�#�#$




� � ��

Hankel,

� ��

Vandermonde,

� ��

Polyn, Vand. � � � � � � ��

Cauchy,

� � � ��

Pick,

� � � � � � � ��



Recent interest

Conferences

Publishing projects

Software project


Recent interest

ConferencesSanta Barbara-1996, Cortona-97, Boulder-1999, Cortona-2000, SouthHadley-2001, ???? Banff-2003 ???.

Publishing projects

Software project


Recent interest


Publishing projects

Edited by Kailath & Sayed, SIAM, 1999.

Edited by Bini, Tyrtyshnikov and Yalamov, Nova Publications, 2000.

Two-Volume set edited by V.O. to AMS series Contemporary Mathematics,November 2001.

2002. A special issue of LAA edited by Dewilde, Sayed & V.O., March 2002.

2002.An AMS/SIAM volume on Fast Algorithms edited by V.O.

2002. A special issue of LAA edited by D.Bini, G.Heinig and E.Tyrtyshnikov.

Software project


Recent interest


Publishing projects

Edited by Kailath & Sayed, SIAM, 1999.

Edited by Bini, Tyrtyshnikov and Yalamov, Nova Publications, 2000.

Two-Volume set edited by V.O. to AMS series Contemporary Mathematics,November 2001.

2002. A special issue of LAA edited by Dewilde, Sayed & V.O., March 2002.

2002.An AMS/SIAM volume on Fast Algorithms edited by V.O.

2002. A special issue of LAA edited by D.Bini, G.Heinig and E.Tyrtyshnikov.

Software project

SLAP (SLAPACK). ????


Why?


Why? Full research circle.Applications. .

Mathematics..

Scientific Computing.



Estimation, Signal processing, Filtering, Circuit simulations, Systems andControl, Image Processing, Error-correcting codes.

Mathematics..





Mathematics..Various interpolation and Approximation problems, Orthogonal polynomials,Moment problems, RKHS, Commutant-lifting theory, etc.






Scientific Computing.Software design. Fast Algorithms for structured matrices. A problem: speedvs. accuracy.





Scientific Computing.Software design. Fast Algorithms for structured matrices. A problem: speedvs. accuracy.

A RECENT DISCOVERY: Though fast algorithms were believed to betypically inaccurate, it turns out that there are better modified algorithms that BLENDSPEED AND ACCURACY. Examples.


Linear predictionIn 1940 N.Wiener won a $2,350 NDRC project on “a design of a predictionapparatus in which, when one member follows the track of an airplane, anotheranticipates where the airplane is to be after a fixed lapse of time.”

If a stochastic process

-.0/ 12/ 34 2 is stationary in the wide sense, theleast-square criterion to find the prediction,

. * 5 )as

.76 � 8 ). )�:9 9 9 � 8 * . *;

leads to the Yule-Walker linear equations:<=>=>=>=>=?=@=?=?=@=?=?AB6 B � ) , , , , , , B �*B ) B6 B � ) ...

.... . .

. . .. . .

......

. . . B6 B � )B * , , , , , , B ) B6CD>D>D>D>D?D@D?D?D@D?D?E<=>=>=>=?=@=?=?=@=FA�8 )8G

...8 *CD>D>D>D?D@D?D?D@DFE

H<=>=>=>=?=@=?=?=@=FAI��

... �CD>D>D>D?D@D?D?D@DFE

9(1)

In the 1949 Wiener monograph there is an Appendix by N.Levinson with a

J K ! GL

algorithm to solve (1). Faster than general

J K ! ML .Fast algorithm

Back to Full Circle. . Potpourri on(Fast Algorithms for)Structured Matrices – p.7/80

Caratheodory-Toeplitz problemGiven: ! � �

complex numbers

- B/ 1

Find:

N K �L H B6 � B ) � � , , ,� B * � * � J KO �O * 5 )L

, analytic on the unit diskP

and positivereal:

Q 8 N K �L R �

in

P

.

Solution: the classical Schur 1917 algorithm:

Matrix interpretation: Computing the Cholesky factorization

S H T , T � for

S H<=?=?=@=?=>=>=>=>=>=?=FAB6 � B �6 B � ) , , , , , , B �*B ) B6 � B �6 B � ) ...

.... . .

. . .. . .

......

. . . B6 � B �6 B � )B+* , , , , , , B ) B6 � B �6CD?D?D@D?D>D>D>D>D>D?DFE

(

S R �

is the solvability condition).

The Cholesky factorization requires

J K ! ML flops. The Schur algorithm usesonly

J K ! GL flops. Fast algorithm.

Back to Full Circle.


Speed vs accuracySpeed:

Standard algorithms Gaussian Elimination

J K ! ML flops

Fast algorithms The Levinson and the Schur

J K ! GLSuperfast algorithms The Brent-Gustafson-Yan

J K ! UWV X G !LAccuracy? Some algorithms are fast AND accurate. One example: FFT. However,too many fast algorithms are fast ways of computing garbage.

Good news. In the last few years a number of fast and accurate algorithms havebeen designed.

Bojanczyk, Brent,de Hoog, and D. R. Sweet (1995). The Schur algorithm forp.d. Toeplitz matrices is backward stable.

Gohberg, Kailath and O. (1995). A reliable algorithm for indefiniteToeplitz(+Hankel)-like matrices.

Also: I.Koltracht, S.Chandrasekaran, M.Gu, P.Van Dooren, M.Stewart, G.Heinig,M.Van Barel, L.Reichel,

Back to Full Circle. Back to ContentsPotpourri on(Fast Algorithms for)Structured Matrices – p.9/80

Floating point arithmetic

Y[Z H -\ ; ] 1 Y[Z H �9 ^ ) ^G ^ M ^`_ ^0a ^0b ^`c ^0de fg hi , � � jIf

Ylk H Nm - k 1 is a floating point representation of k thenO k � YkOO kO n oe fg h

unit roundoff

p � �4 c

in single precision.

Conclusion: We cannot expect more than 7 - 8 correct digits in the solutionscomputed in single precision.


Forward error.We used several fast

J K ! G L algorithms (from Boros-Kailath-O., LAA 1999) to solve a

linear system

q k H ^

with the Hilbert coefficient matrix

q H r )�5 � 4 ) s .




linear system

q k H ^


q H r )�5 � 4 ) s .

K q � t qL K k � t kL H ^ u vWwx t kk ny K qL t qq , K7z{ | 8 } ~ � !� L ;




linear system

q k H ^


q H r )�5 � 4 ) s .n 5 10 15 20 25y 2 K SL

5e+05 2e+13 2e+17 8e+17 1e+19

0 5 10 15 2010−8

10−6

10−4

10−2

100

102

104

SIZE

RE

LATI

VE

ER

RO

R

BKO+mon

BKO+PPP

GECP


Remarkable forward error boundsTHM. (Boros-Kailath-O., LAA 1999). Let

Yk be a solution tor )�� 4 �� s k H N9

computed via an algorithm from Boros-Kailath-O, LAA, 1999. If

�* �9 9 9 � � ) � �) �9 9 9 � �* ;(1)

and the RHS

N

is sign-oscillating then we have :O k � YkOO kO n� ! o � � K oG L remarkable componentwise FORWARD accuracy

Why?

Gohberg-Koltracht (1993) The structured condition numbers of Cauchy andVandermonde matrices are much less than their condition numbers.

Relevant and similar examples of accurate structure-exploiting algorithms: (i) FFT;(ii) the Björck-Pereyra algorithm (1970); (iii) Inversion of Cauchy matrices.

Back to ContentsPotpourri on(Fast Algorithms for)Structured Matrices – p.12/80

How bad are Hankel matrices

� H<=>=>=?=@=?=?=@=?=>=?A~6 ~ ) ~G , , , ~ * 4 )~ ) ~G . .

. ...~G . .. ~G * 4 M

... . .. ~G * 4 M ~G * 4 G )~ * 4 ) , , , ~G * 4 M ~G * 4 G ~G * 4 )

CD>D>D?D@D?D?D@D?D>D?ER �

Conditioning

Tyrtyshnikov (1994):

� & � � � � � & � � � � & � � !� �

No forward accuracy should be expected. What aboutBackward accuracy?


AlgorithmsSlow: Gaussian elimination. Computes

YT p T

in

� H T T �Cost:

J K ! ML .

Stability:

� � � YT YT � � G � J K oL � � � G . Here o is the machine precision,p � �4 )b

.



YT p T

in

� H T T �Cost:

J K ! ML .

Stability:


.

Fast: (There are many...)

Cost:

J K ! GL or

J K ! UWV XG !L .

Stability: ? .



YT p T

in

� H T T �Cost:

J K ! ML .

Stability:


.


Cost:

J K ! GL or

J K ! UWV XG !L .

Stability: ? .

Fast AND Accurate. O.-Michael Stewart-2001:

Cost:

J K ! GL .

Stability:

� � � YT YT � � G � J K oL � � � G .



YT p T

in

� H T T �Cost:

J K ! ML .

Stability:


.


Cost:

J K ! GL or

J K ! UWV XG !L .

Stability: ? .

Fast AND Accurate. O.-Michael Stewart-2001:

Cost:

J K ! GL .

Stability:

� � � YT YT � � G � J K oL � � � G .

HOW ?Potpourri on(Fast Algorithms for)Structured Matrices – p.14/80

Displacement structureKailath, Kung, Morf (1979), Morf (1974), Sakhnovich (1976), Heinig-Rost (1984), Dym(RKHS), Ball-Gohberg-Rodman (interpolation), etc.

Toeplitz- like

� Displacement rank of

�g h e f

rank

K� � � �� L � � !Hankel- like

�

rank

K� � � �� L � � !

Vandermonde- like

�

rank

K� 4 )� � � �� L � � !

Cauchy- like

�

rank

K � � � � �� L � � !

(Below we introduce the more general Confluent Cauchy-like matrices.)

Applications to tangential interpolation

� � H �* K �L H<=>=?=@=?=>=>=>=?=@=FA��

� � . . ....

. . .. . .

. . .� , , , � � �CD>D?D@D?D>D>D>D?D@DFE

; � � H �� X K �L9


Why the names?The displacement rank of a Cauchy matrix

S H r )�� 4 �� s

is equal to 1:

� � S � S� � H r �� 4 �� s � r �� 4 �� s H r �� 4 �� 4 �� s H<=>=>=>=?=@=FA� � , , , �� , , , �

......

...� � , , , �CD>D>D>D?D@DFE

If the displ. rank

K �L � � ! then

�

is called Cauchy-like.

The Pick matrices

� H � )4 �� 5 � �� are Cauchy-like (displ. rank H &

):

� � � � �� H � � � N �N �� H<=?=@=?=>=>=FA� � N )� � NG

......� � N*

CD?D@D?D>D>DFE<A � � , , , �N � ) N �G , , , N �*CE

e f g h

Short generators

9


The uniform way to capture the structureIf the displ. rank of

�

is � then one can factor (nonuniquely)

q¡ � � � q£¢ H ¤ , S He f g h¥

, ;

Thus,

�

is defined by only

& � ! entries of its generator - ¤; S 1

.

Hankel generator:

� � � �� H:<=@=?=>=>=FA� � ~6 � ~ ) � ~G~6 ~ ) 0~G

CD@D?D>D>DFE H<=@=?=>=>=FA~6 �� ~6� ~ )� ~GCD@D?D>D>DFE

<A � � �� CE <A ~6 � � �� ~6 ~ ) ~GCE H

Unfortunately the Hankel structure is destroyed during elementarycomputations.


Composite matricesComposite matrices: Inverses of Hankel matrices,

�4 )

, their products

� ) �G ,Schur complements

�G G � � )G �4 )) ) �G )are NOT Hankel matrices themselves.

If

- � �1 are “structured”, we would like the same to be true for thecomposite matrices.

“Theorem”: The composite matrices will also be structured, if “structure” is takenas DISPLACEMENT STRUCTURE.

Displ. rank

K �L n &

Displ. rank

K �4 )L n &

.

Displ. rank

� ) �G n ¦

.

Displ. rank

K �G G � �G ) �4 )) ) � )G L n &.

The displacement approach to a problem is to solve it

in terms of generator matrices

- ¤; S 1;

and not in terms of the entries of

�

.


Why it is fast?Gaussian elimination Fast Displacement�G H � § )¨G G � ) ©Wªm ) « ) ¤G H ¬¤ ) � )©Wªm )^ )

H � H �

' K ! � �L G � � K ! � �L

Cost:

�¯® & �

.

What about accuracy?


How to insure stability?

� � � �� H q � q �; ° vWw± w � H <A � � �� CE

Generator

q

is non-unique:q³² � q´ ; where

´ �´ � H �9GOOD

´

:

Scale

q² � q <A µ �� ) ©CE so that the new columns of

q

have the same

norm.

Put

q² � q <A B � zz BCEe fg h¶· ¸ ¹ ¶º ¶ *» ¼

in a proper form

<=>=>=?=@=?=FA� ��

......� �

CD>D>D?D@D?DFE

This algorithm is backward stable:

� � � YT YT � � n J K oL � � �

. O.-Stewart-2001.


Superfast displacement algorithms

Confluent Cauchy-like matrices.

LU factorization. Solving a linear system.

Matrix-vector product.


Confluent Cauchy-like matrices

Definition:

confluent Cauchy-like rank

K q½ � � � q¾¢ L � � !Matrices

q ¢ ; q¡ are assumed to be in the Jordan form.

T-like, H-like, V-like, C-like matrices are special cases of confluent Cauchy-likematrices.

We designed a new superfast algorithm for computing

� H T¿

and for solving alinear system with Confluent Cauchy-like matrix

�

.


Cost

± � x À K q¡ � � � q£¢ L � � !9- | );9 9 9 ; |�· 1

are the sizes of the Jordan blocks of

q .- } );9 9 9 ; }ÂÁ 1


q ¢ .

Unified Algorithm:Ã K !L

confl. Cauchy-like

J K ! UWV XG ! , � � � Ä ·/ 3 ) ÅÆ* UWV X * ÅÆ � Ä Á/ 3 ) ¸Æ* UWV X * ¸Æ �L

Includes as special cases:Toeplitz-like

J K ! UWV XG !L

Hankel-like

J K ! UWV XG !L

Vand.-like

J K ! UWV X M !L

Cauchy-like

J K ! UWV X M !L

An important ingredient of this unified algorithms is computing

matrix-vector product .


Fast algorithms. Some references

Martin Morf’s breakthrough thesis (Stanford, 1974). First fast

J K ! GL Schur-typeand Levinson-type algorithms for Toeplitz-like matrices.

Monograph by Heinig and Rost (1984). First fast

J K ! GL Levinson-type algorithms

for Vandermonde- like and Cauchy- like matrices.

Survey by Kailath and Sayed (1995). FastJ K ! GL Schur-type algorithms for basic

classes with displacement structure.

Survey by V.O. (1997) Numerical accuracy issues. Fast

J K ! GL algorithms for

confluent Cauchy-like matrices in the context of tangential interpolation .


Superfast algorithms

Morf’s (1974)’s thesis. A sketch of a superfast algorithm for Toeplitz-like matrices(not clean?).

Brent-Gustafson-Yan (1979) First

J K ! UWV X G !L algorithm for solving Toeplitz (not-like ) linear equations.

Bitmead-Anderson (1980) and Morf (1980). General Toeplitz- like case,J K ! UWV XG !L .

Gohberg-O. (1994) : Cauchy-like and Vandermonde-like matrices.

Our superfast algorithmApplies to the most general confluent Cauchy-like displacement structure .


MV product. Classical algorithms.

DFT matrices (i.e., Vandermonde

matrices with special nodes) Fast Fourier transform

Ã K !LToeplitz/Hankel matrices convolution

Ã K !LVandermonde matrices multipoint

polynomial evaluation

Ã K !L UWV X !

inverse Vandermonde matrices polynomial interpolation

Ã K !L UWV X !

Cauchy matrices multipoint

rational evaluation

Ã K !L UWV X !

inverse Cauchy matrices rational interpolation

Ã K !L UWV X !

Ã K !L H ÇÉÈÊ ! UWV X ! if the field

Ë

supports FFT’s of length !! UWV X ! UWV X UWV X ! otherwise


Cost of matrix-vector product (O.-Shokrollahi (2000))

Thm. Confluent Cauchy-like matrices can be multiplied by vectors using

S K !L H J K ! UWV X ! , � � � Á�3 ) | �! UWV X !| ��

¸� 3 )

Ì � ! UWV X !Ì � �L

flops, where- | );9 9 9 ; |Í· 1


q .- } );9 9 9 ; }ÂÁ 1


q ¢ .S K !L can vary from

J K ! UWV X !L to

J K ! UWV X G !L depending upon the Jordanstructure.

Back to contents


The classical scalar Nevanlinna-Pick problem

Given:points � );9 9 9 ; � * in the RHP:

Qw �/ Î �values

N );9 9 9 ; N* in the unit disk:

O N/ O � �Construct an interpolant

N K �/ L H N/ , such that

(i)

N K �L is passive O N K �L O � �in the RHP

(ii)

N K �L is rational


First papers and results

Solvability condition Pick (1916):

� H � )4 �� 5 � �� R �Cost: The Nevanlinna (1929):

J K ! GL operations.

Recall:

Standard algorithmsJ K ! ML arithm. operations

Fast algorithmsJ K ! GL

Superfast algorithms

J K ! UWV X M !L or

J K ! UWV X G !L

Is the Nevanlinna-Pick algortithm fast?


The algorithm is FAST

The Nevanlinna algorithm Its matrix interpretation

computation of the interpolant

N K �L The Cholesky factorization� H T T �, [

Tis lower triangular] for the Pick

matrix

� H � )4 �� 5 � ��

Arithmetic complexityAlgorithm Cost

The standard Cholesky

J K ! ML

The Nevanlinna algorithm

J K ! GL

We shall show how to do faster than

ÏÑÐ ÒÓ


Applications to system and circuit theory

Many physical phenomena can be described by a Linear Time-Invariant system:

« K �L Ô Õ K �L Ô � K �L H Õ K �L , « K �L

Passivity

Ö O Õ K �L O n � ÖConservation of energy

Energy of input « K �L ÎEnergy of output � K �L .

Many engineering applications have been discovered, e.g., the classicalNevanlinna (1929) algorithm is known under the name “Darlington (1939)synthesis procedure”


Tangential interpolationMIMO systems (multi-input-multi-output) give rise to matrix rational functions× K �L H r�Ø � � § � ¨Ù� � § � ¨ s

.

�)�G�Ú

ÔÔ

...

Ô× K �L

� )�G�Û

ÔÔ...

Ô

This gives rise to the tangential interpolation conditions:r �) , , , �Ú s , × K �/ L H r � ) , , , � Û s


Confluent interpolation conditions

Given: Now for each interpolation point �/ we are given two

chains of vectors:

- �/ );9 9 9 ; �/ÝÜ ÅÆ 1 - �/ );9 9 9 ; �/ÝÜ ÅÆ 1

We consider confluent interpolation conditions:

r �/ )9 9 9 �/ÝÜ ÅÆ sßÞ<=>=>=?=@=?=@=?=?=@=?=?Aà K �/ L à á K �/ L , , , , , , â ã³ä Æå ªæ § �Æ ¨§ ÅÆ 4 )¨ç� à K �/ L . . .

...� � . . .. . .

......

. . .. . . à á K �/ L� , , , , , , � à K �/ L

CD>D>D?D@D?D@D?D?D@D?D?EH

r �/ÝÜ ) , , , �/ÝÜ ÅÆ s9


Two special casesNevanlinna-Pick: �/ , à K �/ L H �/

Caratheodory-Fejer, i.e., rational passive Hermite-type interpolation:

r � � 9 9 9 � sèÞ<=>=?=@=?=?=@=?=>=>=>=?=@=?=FAà K �/ L à á K �/ L , , , , , , )§ ÅÆ 4 )¨ç à § ÅÆ 4 )¨ K �/ L

� à K �/ L . . ....� � . . .

. . ....

.... . .

. . . à á K �/ L� , , , , , , � à K �/ L

CD>D?D@D?D?D@D?D>D>D>D?D@D?DFEH

r �/ÝÜ ) , , , �/ÝÜ ÅÆ s9

or in the other words

à K �/ L H �/ ); à á K �/ L H �/ G; �& é à á á K �/ L H �/ M;9 9 9


Solutions and fast

ê

algorithmsScalar case : see, e.g., the monographs by Akhiezer (1965), Krein-Nudelman

(1977) and Rosenblum and Rovnyak (1985).

Fedchina (1972): Tangential problem without confluencies (via AAK).

Kovalishina and Potapov (1974) - direct

J K ! G L algorithm for tangential problem(without confluences). Via Schur-type alg. of Potapov (1966).

Fedchina (1975): First

J K ! GL algorithm for Confluent tangential problem.

Many deep results have been obtained using different approaches:

The band-extension-method by H.Dym, I.Gohberg.

The Buerling-Lax-theorem approach by J.Ball, J.Helton.

The commutant-lifting-theorem approach by C.Foias, A.Frazho.

The reproducing-kernel-Hilbert-spaces approach by H.Dym.

The state-space approach by Van Dooren, J.Ball, I.Gohberg, L.Rodman.

The discrete-transmission-lines approach by Lev Ari, T.Kailath.

The diagonal-calculus approach by P.Dewilde.

Nudelman, Delsarte-Genin-Kamp, Sarason, Sz-Nagy-Foias, Sakhnovich,Rosenblum-Rovniak, Kimura, Limebeer-Anderson, Katsnelson, Kirstein, Fritzsche,Dijksma, Langer, Sayed-Kailath-Lev Ari-Constantinescu, Alpay, Bolotnikov


Computational cost of our problem

Given:

- �/ 1 ·/ 3 ) are given ë points- |/ 1 ·/ 3 ) are their multiplicities- �/ );9 9 9 ; �/ÝÜ ÅÆ 1 ·/ 3 ) are the associated chains- �/ );9 9 9 ; �/ÝÜ ÅÆ 1 ·/ 3 )

The cost

S K !L is (O.-Shokrollahi (1999)):

S K !L H J K Ã K !L , UWV X ! , � � � Ä ·/ 3 ) ÅÆ* UWV X * ÅÆ �L

where ! H | )�9 9 9 � |�·

Ã K !L HÇíìîì[Èìlì[Ê! UWV X ! if the field

Ë

supports FFT’s

of length !! UWV X ! UWV X UWV X ! otherwise


Two simplest cases (with FFT)

One point. Tangential Caratheodory-Fejer. If ë H �

(so ! H | )) then

S K !L H J K ! UWV XG !Ln points. Tangential Nevanlinna-Pick. If ë H ! then

S K !L H J K ! UWV X M !L

Compares favorably with the earlier

J K ! GL bounds.

Back to contentsPotpourri on(Fast Algorithms for)Structured Matrices – p.37/80

CodesCodes are used when messages are to be transmitted over noisy channels.

E S S A GM E S S A GM EA

E S S A GM E

E S S A GM E

E S S A GM E

E S S A GM E

E S S A GM E

S S A GM EA

With coding

Without coding


Redundnancy. Linear codes

/g h e fï ï ï ï ï ï ï ï ïe f g h*

A

Ì

-dimensional subspace of

ð* Ùis called an

� !; Ì � Ù-code.Its elements are called codewords.

The minimum distance is the the minimum distance between two distinctcodewords.

Minimum distance H minimum distance to the zero


Error correctionA code of minimum distance

µ

is capable of correcting up to 8ñ H K µ � �Lò &errors.

Hamming balls ofradius e

e

d


Two basic problems in coding

Design a good code, i.e., with a large minimum distance.

Design fast decoding algorithms.


Reed-Solomon CodesGiven: �);9 9 9 ; �* distinct elements of

ð Ù9 .

Encode

- k );9 9 9 ; k/ 1îó ô õ N K �)L ;9 9 9 ; N K �* L öwhere

Ì n !

where

N K �L H k )� kG ��:9 9 9 � k/ �/ 4 );


Excellent minimal distance of the RS Codes

The above code has dimension

Ì

.

What about minimum distance? (distance to zero)Theorem. Nonzero polynomial of degree

K Ì � �L

over a field can have at most| K Ì � �L

roots in the field.Implication:

the above code has minimum distance ! � K Ì � �L.

This is excellent: Increasing ! by

&

increases the minimum distance by

�

.

What about decoding algorithms?


Encoding

N K �/ L H k )� kG �/ � k M �G/ �9 9 9 � k/ �/ 4 )

<=@=?=?=@=?=?A� �) �G ) , , , �/ 4 ))� �G �G÷G , , , �/ 4 )G

......

......� �* �G * , , , �/ 4 )*

CD@D?D?D@D?D?E<=>=>=?=@=?=>=>=>=FAk6 k )kG

...k/ 4 )CD>D>D?D@D?D>D>D>DFE

H<=@=?=?=@=?=?AN K �)LN K �G L

...N K �* LCD@D?D?D@D?D?E9

Decoding?


Welch-Berlekamp Method

Let

K � );9 9 9 ; �* L have distance

n K ! � ÌLò &

from codeword

õ N K �)L ;9 9 9 ; N K �* L ö;Nø ð Ù�. �úù/ .Find � ø ð Ù�. �£ù § * 5/ ¨û G and

~ø ð Ù�. �£ù § * 4/ ¨û G such that

ü � H �;9 9 9 ; !ñ ~ K � �L � �� K ��L H �9 Then

N K. L H � K. L~ K. L9

vanishes at these pointsh

sent word

received word

f

Error locator polynomial


Matrix interpretation

! H �

and

Ì H ¦

:

<=?=?=@=?=>=>=>=>=>=?=@=?=FA� �) �G ) �M ) �_ ) � ) � ) �)� �G �GýG �MG �_G �G �G �G� �M �G M �MýM �_ M � M � M �M� �_ �G _ �M_ �_ý_ �_ �_ �_� �a �G a �Ma �_ a �a �a �a� �b �G b �Mb �_ýb �b �b �bCD?D?D@D?D>D>D>D>D>D?D@D?DFE

e f g hDispl. rank 2

<=?=@=?=?=@=?=>=>=>=>=>=?=@=?=>=>=FA�6 � )� G� M� _ ~6 ~G

CD?D@D?D?D@D?D>D>D>D>D>D?D@D?D>D>DFEH �9


RS codes via displacement

<=>=>=FA)� ª

. . . )�þCD>D>DFE<=>=>=FA� �) �G ) �M ) �_ ) � ) � ) �)

......

...� �b �G b �Mb �_ýb �b �b �bCD>D>DFE �

<=?=@=FA� �) �G ) �M ) �_ ) � ) � ) �)

......

...� �b �G b �Mb �_ÿb �b �b �bCD?D@DFE

<=@=?=>=>=>=?=@=?=>=>=>=?=?A� � � � � � �� CD@D?D>D>D>D?D@D?D>D>D>D?D?E

H

<=>=?=FA*

� � � � �

*

�

......

......

...

*

� � � � �

*

�CD>D?DFE

e f g h

rank 2


List-Decoding of RS Codes

What if the number of errors is larger than

K ! � ÌLò &

?


Bivariate InterpolationGiven: Points

K �); � )L ;9 9 9 ; K �* ; �Í* L over a field

Ë

, and integers

� n µ6 n µ ) n , , , n µ�� .Want: Nontrivial Polynomial

� K �; �L ñ H Ä � �36 ~ �K �L � �

with

� w X K ~ �L n µ �, such that� K � �; � �L H �

for

� H �;9 9 9 ; !.Existence?

N

exists, if

Ä � �36 K µ �� L R !.�ñ H ~6 6 � ~6 ) �� ~6 G �G � ~ )6 � � ~ ) ) �� ~G6 �G9We have

��

� �) �G ) � ) � ) �) �G )� �G �G÷G �G �G �G �GG� �M �G M � M � M �M �G M� �_ �G _ �_ �_ �_ �G_� �a �G a �a �a �a �Ga�

��

��~6 6 ~6 )~6 G~ )6 ~ ) )~G6�

��H �9


List decoding via displacement (O.-Shokrollahi)<=>=?=@=?=FA� �)

. . . � �aCD>D?D@D?DFE

<=>=>=?=@=?=@=FA� �) �G ) � ) � ) �) �G )� �G �G÷G �G �G �G �GG� �M �G M � M � M �M �G M� �_ �G _ �_ �_ �_ �G_� �a �G a �a �a �a �GaCD>D>D?D@D?D@DFE

�<=?=@=?=>=>=>=FA� �) �G ) � ) � ) �) �G )� �G �GýG �G �G �G �GG� �M �G M � M � M �M �G M� �_ �G _ �_ �_ �_ �G_� �a �G a �a �a �a �GaCD?D@D?D>D>D>DFE

<=?=@=?=?=@=?=>=>=>=FA� � � � � �� CD?D@D?D?D@D?D>D>D>DFE

H<=>=>=?=@=?=?=>A�ò �) � )ò �)� �G ) �G )ò )� � ) �)�ò �G �Gò �G � �GÿG �GGò G � �G �G�ò �M � Mò �M � �G M �G Mò M � � M �M�ò �_ �_ò �_ � �G _ �G_ò _ � �_ �_�ò �a �aò �a � �G a �Gaò a � �a �aCD>D>D?D@D?D?D>E

<=>=FA� � � � � �� CD>DFE9


Appendix 1.Stability test for matrix polynomials

The classical Schur-Cohn test needs

J K ! G L to check discrete-time stability (rootsin the unit circle) of a polynomial

m K �L with

� w X m K �L H !.Matrix polynomals? T K �L H � * � � � * 4 ) q * 4 )�9 9 9 � � q )� q6 ;

where

q/ are |Þ | matrices.

The stability of

T K �L can be checked in

J K ! m{ � G !L operations, provided | issmall.


Appendix 2.Schur-type algorithm for Hessenberg displacementstructure

Let

rank

K q¢ � � � q � L � � !In [K99] it is claimed: “The restriction that

- q ¢ ; q¡ 1are lower triangular is the most

general one that allows recursive triangular factorization (cf. a result in [LK86]).”

Heinig-O. In fact,

�

can be recursively factored as� H T¿

also in cases- q¢ ; q½ 1

are non-triangular, e.g., Hessenberg. Cost: ! G operations.


Appendix 3.How bad are Pick matrices?

Bad news: The cond.number of real p.d. Pick matrices

�growths exponentially

with its size !.Good news: Pick matrices � H r ©� 5 ©�� 5 � � scan be effectively preconditioned by the Cauchy matrices

S H r )�� 5 � � s

generated by the same nodes.


Appendix 3.Esponential ill-conditionig

THM:

Pick matrices are BAD: If

� H r ©� 5 ©�� 5 � � s R �theny G K �L R ë * 4 )

.

Not very bad, though: ë H � � ª 5 � �� ª4 �� G R �.


Appendix 3.A numerical illustration. Cauchy matrices

0 5 10 15 20100

102

104

106

108

n

(d)

0 5 10 15 20100

102

104

106

108 (a)

n0 5 10 15 20

1

2

3

4

5

6

n

(b)

0 5 10 15 20100

101

n

(c)

Figure 0: (a)y G K �* L , (b)y G K �* Lò y G K �* 4 )L

, (c)

Ky G K �* L L )û *

,(c) The optimal nodes. Potpourri on(Fast Algorithms for)Structured Matrices – p.55/80

Appendix 3.Gram matrices

Let we have a certain inner product

� N K �L ; � K �L �

, and a space of functionsspanned by

- ë )K �L ; ëG K �L ;9 9 9 ; ë * K �L 1

. Then

� */ 3 ) k/ ë/ K �L ; */ 3 ) ^/ ë/ K �L � H r ^ )9 9 9 ^ * s Ã

<=@=?=FAk )

...k*CD@D?DFE;

where

Ã H r � ë/ K �L ; ë � K �L � sis called the Gram matrix (moment matrix).

Hankel matrices are the Gram matrices for ë/ K �L H �/

and� N K �L ; � K �L � H � �» N K �L � K �L � K �L µ �, where � K �L

is a certain generating function.

Toeplitz matrices are the Gram matrices for ë/ K �L H �/

and� N K �L ; � K �L � H � G ¢6 N K 8 ��L � K 8 ��L � K 8 ��L µ�

, where � K �L

is a certain generatingfunction.


Appendix 3.Generating function for Pick matrices

Gram matrices: If

� N; � �� H �» 4 » N K �L � K �L � K �L µ �then

� H r � )� 4 ��Æ ; )�4 �� s

If

� R �

then the above � K �L can be chosen to be a good generating function, i.e.,

� � | n � K �L n Ã

on

�

.


Appendix 3.Cauchy matrices. Constant generating function

If � K �L H B{ !z }

then

� H r � ¶ * Á ¸�� 5 � � s

is a Cauchy matrix.

Example: The famous Hilbert matrix

� H<=>=>=>=?=@=?=>=>=>=?=@=?=?=@=?=>=>=>=FA� )G )M )_ , , ,

)G )M )_ )a)M )_ )a )b

)_ )a )b )c)a )b )c )d

CD>D>D>D?D@D?D>D>D>D?D@D?D?D@D?D>D>D>DFE�/ H �ò & � K Ì � �L

Hilbert matrix is a standard test matrix form numer. algorithms because of itsill-conditioning. Potpourri on(Fast Algorithms for)Structured Matrices – p.58/80

Appendix 3.Preconditioning

If

� � | n � K �L n Ã

is a good generating function for

�, and

Sis a Cauchy

matrix with the same nodes as

�

, then

y G K Ë4 ) �L n Ã|9


Appendix 3.Numerical illustration

5 10 151

1.2

1.4

1.6

1.8

2 (a)

n5 10 15

1

1.2

1.4

1.6

1.8

2 (b)

n

5 10 151

1.2

1.4

1.6

1.8

2 (c)

n5 10 15

1

1.2

1.4

1.6

1.8

2 (d)

n

Figure 0: Eigenvalues of preconditioned Pick matrices. � K �L H � � K �G � �L 4 )

.(a) �/ H &/ 4 )

, (b) �/ H &4/ 5 )

, (c) �/ H & G/ 4 * 4 )

, (d) Optimalnodes.


Appendix 4.Vandermonde matrices. Multipoint polynomial evaluation.

Matrix-vector product:<=>=>=>=>=?=FA� �) �G ) , , , �* 4 ))� �G �G÷G , , , �* 4 )G

......

......� �* �G * , , , �* 4 )*

CD>D>D>D>D?DFE<=>=>=>=>=?=FAk6 k )

...k* 4 )CD>D>D>D>D?DFE H<=>=>=>=>=?=FAN K �)LN K �GL

...N K �* LCD>D>D>D>D?DFE

Polynomial multipoint evaluation ofN K �L H k6 � k ) �� kG �G �9 9 9 � k* 4 ) �* 4 )

at �); �G;9 9 9 ; �*9

Matrix-vector product for the inverse of Vandermonde matrix: interpolation.

Discrete Fourier transforms : a special case. �/ H 8 ��K G ¢ �/ * L


Appendix 4Cauchy matrices. Multipoint rational evaluation.

Matrix-vector product:<=>=>=FA)� ª4 � ª , , , )� ª4 ��

......)�� 4 � ª , , , )�� 4 ��

CD>D>DFE<=>=>=FAk6

...k* 4 )CD>D>DFE H<=>=>=FAN K �)L

...N K �* LCD>D>DFE

Rational multipoint evaluation of

N K �L H k6 �� )�9 9 9 � k* 4 ) �� +*9

at �);9 9 9 ; �*

Matrix-vector product for the inverse of Cauchy matrix: rational interpolation.


Appendix 4.Toeplitz matrices. Discrete Convolution.

Polynomial multiplication (discrete convolution):K k6 � k )� � kG � G �9 9 9 L K ^6 � ^ )� � ^G � G �9 9 9 L H

k6 ^6 � K k )^6 � k6 ^ )L � � K kG ^6 � k )^ )� k6 ^ ML � G �9 9 9

Matrix-vector product:<=?=@=?=?=@=?=>=>=?Ak6 k ) k6kG k ) k6

.... . .

. . .. . ., , , kG k ) k6

CD?D@D?D?D@D?D>D>D?E<=?=@=?=?=@=?=>=>=?A^6 ^ )^G

...

CD?D@D?D?D@D?D>D>D?EH

<=?=@=?=?=@=?=>=>=?Ak6 ^6k )^6 � k6 ^ )kG ^6 � k )^ )� k6 ^ M

...

CD?D@D?D?D@D?D>D>D?E


Appendix 5.A “challenging” interpolation problem

Given:

& ! pairwise different numbers

- z );9 9 9 ; z * ; } );9 9 9 ; } * 1.

Construct: a rational scalar function

� K �L

such that� K z �L H �

and z �’s are the only zeros of

� K �L

.�4 )K } �L H �

and

} �’s are the only poles of

� K �L.N K �L H �

.

?


Appendix 5.A solution via a structured matrix

� K �L H � � r � , , , � s <=>=>=FA)�4 ¸ ª

. . . )�4 ¸�CD>D>DFE<=>=>=FA)Á ª4 ¸ ª , , , )Á ª4 ¸�

......)Á � 4 ¸ ª , , , )Á � 4 ¸�

CD>D>DFE4 ) <=>=>=FA�

... �CD>D>DFE

Clearly

} �’s are the only poles of

� K �L

.

Clearly

N K �L H �

.

� K z/ L H � � r )Á Æ 4 ¸ ª , , , )Á Æ 4 ¸� s <=>=>=FA)Á ª4 ¸ ª , , , )Á ª4 ¸�

......)Á � 4 ¸ ª , , , )Á � 4 ¸�

CD>D>DFE4 ) <=>=>=FA�

... �CD>D>DFE H �9

z �’s are the only zeros of� K �L

.


Appendix 5.Interpolation and Structured matrices

Remark that in the above simplest scalar rational interpolation problem� K z �L H �

.�4 )K } �L H �

.

the Cauchy matrix appears.

It turns out that varous structured matrices naturally appear in a number ofinterpolation problems.


Appendix 5 .State-space formulas for

� � � �

and structured matrices!

Many interpolation problems have state space solutions of the form

� � � �

= � + S¢ � � " # $&% � � � �4 ) ¤

All the interpolation data are captured by- S¢ ; q£¢ ; q¡ ; ¤ 1

and

�

is determinedfrom the displacement equation

� q£¢ # q¡ � � ¤ S¢


Appendix 5.Homogenuous tangential interpolation problem

Given:

points �);9 9 9 ; �* ø '

points � );9 9 9 ; �+* ø '

rows

^ );9 9 9 ; ^ * ø ' )( ¥

columns B );9 9 9 ; B * ø ' ¥( )

Construct a rational �Þ � matrix function

) K �L with value� ¥at infinity such that

1.The zeros of

� w u ) K �L occur only at

- �/ 1and

the zeros of

� w u )4 )K �L occur only at- �/ 1

.

2. Left int. cond.:

� w u ) K �L has a simple zero at �/ and

^/ , ) K �/ L H �

3. Right int. cond.:

� w u )4 )K �L has a simple zero at �/ and

)4 )K �/ L , B/ H �


Appendix 5.Solution as given in BGR(1990)

) K �L H � ¥� S¢ K � �* � q¾¢ L 4 ) �4 ) ¤ where q¾¢ H �� X r � )9 9 9 �* s; q½ H �� X r �)9 9 9 �* s;

S¢ H r B ) , , , B�* s; ¤ H<=@=?=FA^ )

...^ *CD@D?DFE

and

� H r ��* � �� 4 �� s

is obtained from

q½ � � � q¾¢ H ¤ , S¢

Back to Displacement Back to uniform


Appendix 5.Two realizations

Right int.c. for

)4 )

define

- S¢ ; q£¢ 1

in

) K �L H � � S¢ K � � � q£¢ L 4 ) ¤¢(*).

Left int.c. for

)

define

- q ; ¤ 1

in

)4 )K �L H � � S K � � � q L 4 ) ¤ (**) .

Invert (*):

) K �L H � � S¢ K � � � q (¢ L 4 ) ¤¢ whereq (¢ H q¢ � ¤¢ S¢

(***).

Then (**) and (***) are similar:S¢ H S �; ¤¢ H �4 ) ¤ ; � q (¢ H q½ �

Hence

� q¢ � q½ � H � ¤¢ S¢ H ¤ S¢ and

) K �L H � � S¢ K � � � q¾¢ L 4 ) �4 ) ¤ 9


Appendix 5.Symmetries

If

× K �L � � × K �L H �

on

� �

then ×4 ) H � × K �L � �and hence� � H � �� .S¢ H � ¤ � .

× K �L H � � � ¤ � K � � � q¾¢ L 4 ) �4 ) ¤ 9

where q � � � q � H ¤ , � , ¤ �


Appendix 6.Homogenuous confluent �

-contractive interpolation

Given :points

- �/ 1

inside RHP

Chains of

�Þ +

vectors

- «/ );9 9 9 ; «/ÝÜ ÅÆ 1,

Construct : a rational

+Þ +

matrix function

× K �L s.t.

(i) Tangential homogenuous confluent interpolation conditions:

r «/ )9 9 9 «/ÝÜ ÅÆ s<=>=>=?=@=?=>=>=>=>=>=?=@=?=FA× K �/ L × á K �/ L , , , , , , , ã ä Æå ªæ § �Æ ¨§ ÅÆ 4 )¨ç� × K �/ L . . .

...� � . . .. . .

......

. . .. . . × á K �/ L� , , , , , , � × K �/ L

CD>D>D?D@D?D>D>D>D>D>D?D@D?DFEH r � , , , � s9

(ii)

�-unitary on

� � × K �L , � , × � K �L H �; � ø � ��-contractive inside RHP

× K �L , � , × � K �L n �; Qw � Î �


Appendix 6.BGR-90 solution and confluent Cauchy-like matrices

Solution (Ball-Gohberg-Rodman):

× K �L H � � � ¤ � K � � � q¾¢ L 4 ) �4 ) ¤ 9Solvability condition :

� R �

, where

q ¢ � � � q �¢ H ¤ � ¤ �

q¾¢ H<=>=?=@=?=@=FA� Å ª K � )L � Å- K �GL

. . . � Å�. K �·LCD>D?D@D?D@DFE�

; ¤¢ H<=?=?=@=?=>=>=>=?=@=?=>=>=>=?=@=?=@=?=?=@=?=>=>=>=?=@=?=>=>=FA�) ) � ) )

......�)Ü ÅÆ � )Ü ÅÆ�G ) �G )

......�GÜ ÅÆ �GÜ ÅÆ

......

......�· ) �· )

......�·Ü Å. �·Ü Å.

CD?D?D@D?D>D>D>D?D@D?D>D>D>D?D@D?D@D?D?D@D?D>D>D>D?D@D?D>D>DFE


Appendix 6.Example: Representations of scalar rational functions

State-space form:

× K �L H � � S¢ K � � � q¾¢ L 4 ) �4 ) ¤ H

� �/10g h e fr � , , , � s

§ � 24 3 0 ¨å ªg h e f<=?=>=FA)�4 � ª

. . . )�4 ��CD?D>DFE

�4 )g h e f<=?=>=FA)� ª4 � ª , , , )� ª4 ��

......)�� 4 � ª , , , )�� 4 ��

CD?D>DFE4 ) 465g h e f<=?=>=FA�

... �CD?D>DFE

Factored form:

× K �L H �4 � ª�4 � ª , �4 � -�4 �- ,9 9 9 , �4 � ��4 ��

Complexity of evaluation of

� � � �

State-space form:

J K ! ML flops to invert

�

. 7 Factored form :

J K !L .


Appendix 6.Factorization of rational matrix functions� � � � � � � � � & � � �

First-order

) )K �L .�

-contractive functions.Potapov (1960).Belevitch (1963).Youla (1964).Alpay-Dym (1986).

General functions.Dewilde-Vanderwalle (1975).Dewilde-Van Dooren (1981)Sakhnovich (1976)Bart-Gohberg-Kaashoek-Van Dooren (1979)Gohberg-Olshevsky (1994).

For a superfast divide-and-conquer algorithm we need a higher-order factorization× K �L H ) )K �L , )G K �LPotpourri on(Fast Algorithms for)Structured Matrices – p.75/80

Appendix 6.Divide-and-conquer factorization

× K �L8

)G K �L9) )K �L

8

) )G K �L9) ) )K �L

89

, , ,8 × G K �L9× )K �L

Higher-order

) )K �L (needed for superfast algorithms).

Sakhnovich (1976).

Bart-Gohberg-Kaashoek-Van Dooren (1980), Bart-Gohberg-Kaashoek(1979).

Gohberg-O. (1994). Alpay-Dym (1994), Dym (1996).


Appendix 6.The first factor in

� �� & � � �

does not requirecomputations

Let

× K �L H � � S¢ K � � � q¾¢ L 4 ) �4 )) ¤ H

� � r S¢Ü ) S¢Ü G s K � � � <A q¢Ü ) �: q¢Ü G

CEL 4 ) <A � ) ) � )G�G ) �G GCE 4 ) <A ¤ Ü )¤ Ü GCE

Then

) )K �L H � � S¢Ü ) , K � � Ú ª� q¢Ü )L 4 ) , � ) )4 ) , ¤ Ü );

What about & � � �

?


Appendix 6.Sakhnovich’s and better formulas for& � � � � " ; � K &L% � � " # $&% ; & � � � !� �& < K &L=

Sakhnovich formulas

S § G ¨¢ H K S¢Ü ) � )G � S¢Ü G �G GL �4 )G ; ¤ § G ¨ H �G K �G ) ¤ Ü )� �G G ¤ Ü G L

lead to a very slow

J K ! _L algorithm.

The formulas

S § G ¨¢ H S¢Ü G � S¢Ü ) �4 )) ) � )G ¤ § G ¨ H ¤ Ü G � �G ) �4 )) ) ¤ Ü )9

lead to a superfast D&C-algorithm.


Appendix 6.Superfast D&C-factor algorithm

Input:

× )K �L H � � S § )¨¢ K � � � q£¢ L 4 ) �4 )) ¤ § )¨

Output:Minimal factorization

× )K �L H ) )K �L , )G K �L ;The inverse

×4 )) K �L H � � S § )¨ K � � � q¡ L 4 ) � ) ¤ § )¨¢ 9

Steps: 1. Write the first factor

) )K �L H � � S § )¨¢Ü ) , K � � � q¢Ü )§ )¨L 4 ) , �4 )) ) , ¤ § )¨ Ü )9

2. Call the algorithm D&C-Factor to

) )K �L to obtain its inverse

)4 )) K �L .

3. Compute the factor

)G K �L H � � S § G ¨¢ K � � � q¢Ü GL 4 ) �4 )G ¤ § G ¨ 9

4. Apply the algorithm D&C-Factor to)G K �L to obtain its inverse

)4 )G K �L .

5. Multiply the factors in

×4 )) K �L H )4 )G K �L , )4 )) K �L9


The end

Back to Contents


potpourri on (fast algorithms for) structured matricesolshevsky/talks/potpourri_02.pdfthe themes...

Documents