A superfast algorithm for confluent rational tangential interpolation problem via matrix-vector...

Contemporary Mathematics

A Superfast Algorithm for Confluent Rational TangentialInterpolation Problem via Matrix-vector Multiplication for

Confluent Cauchy-like Matrices

Vadim Olshevsky and Amin Shokrollahi

Abstract. Various problems in pure and applied mathematics and engineer-ing can be reformulated as linear algebra problems involving dense structuredmatrices. The structure of these dense matrices is understood in the sensethat their n2 entries can be completeley described by a smaller number O(n)of parameters. Manipulating directly on these parameters allows us to de-sign efficient fast algorithms. One of the most fundamental matrix problemsis that of multiplying a (structured) matrix with a vector. Many fundamen-tal algorithms such as convolution, Fast Fourier Transform, Fast Cosine/SineTransform, and polynomial and rational multipoint evaluation and interpola-tion can be seen as superfast multiplication of a vector by structured matrices(e.g., Toeplitz, DFT, Vandermonde, Cauchy). In this paper, we study a gen-eral class of structured matrices, which we suggest to call confluent Cauchy-likematrices, that contains all the above classes as a special case. We design a newsuperfast algorithm for multiplication of matrices from our class with vectors.Our algorithm can be regarded as a generalization of all the above mentionedfast matrix-vector multiplication algorithms. Though this result is of interestby itself, its study was motivated by the following application. In a recentpaper [18] the authors derived a superfast algorithm for solving the classicaltangential Nevanlinna-Pick problem (rational matrix interpolation with normconstrains). Interpolation problems of Nevanlinna-Pick type appear in severalimportant applications (see, e.g., [4]), and it is desirable to derive efficientalgorithms for several similar problems. Though the method of [18] can beapplied to compute solutions for certain other important interpolation prob-lems (e.g., of Caratheodory-Fejer), the solution for the most general confluenttangential interpolation problems cannot be easily derived from [18]. Derivingnew algorithms requires to design a special fast algorithm to multiply a con-fluent Cauchy-like matrix by a vector. This is precisely what has been done inthis paper.

1991 Mathematics Subject Classification. Primary: 15A06 Secondary: 47N70, 42A70.Key words and phrases. Rational matrix tangential interpolation. Nevanlinna-Pick problem.

Caratheodory-Fejer problem. Cauchy matrices. Superfast algorithms.This work was supported by NSF grant CCR 9732355.

c©0000 (copyright holder)

1

2 VADIM OLSHEVSKY AND AMIN SHOKROLLAHI

Toeplitz, T =[

ti−j

]Hankel, H =

[hi+j−2

].

t0 t−1 · · · · · · t−n+1

t1 t0 t−1

......

. . . . . . . . ....

.... . . t0 t−1

tn−1 · · · · · · t1 t0

h0 h1 h3 · · · hn−1

h1 h2 . . . hn

h2 . . . . . ....

... hn−1 hn h2n−3

hn−1 hn · · · h2n−3 h2n−2

Vandermonde, V =[

xj−1i

]Cauchy, C =

[1

xi−yj

]

1 x1 x21 · · · xn−1

1...

......

...1 xn x2

n · · · xn−1n

1x1−y1

· · · 1x1−yn

......

1xn−y1

· · · 1xn−yn

Figure 1. Some examples of matrices with structure.

Toeplitz matrices convolution M(n)Hankel matrices convolution M(n)Vandermonde matrices multipoint polynomial M(n) log n

evaluationDFT matrices (i.e., Vandermondematrices with special nodes) discrete Fourier transform M(n)inverse Vandermonde matrices polynomial interpolation M(n) log nCauchy matrices multipoint rational M(n) log n

evaluationinverse Cauchy matrices rational interpolation M(n) log n

Figure 2. Connection between fundamental algorithms and struc-tured matrix-vector multiplication

1. Introduction

1.1. Several examples of matrices with structure. Structured matricesare encountered in a surprising variety of areas (e.g., signal and image processing,linear prediction, coding theory, oil exploration, to mention just a few), and algo-rithms (e.g., for Pade approximations, continuous fractions, classical algorithms ofEuclid, Schur, Nevanlinna, Lanzcos, Levinson). There is an extensive literature onstructured matrices, we mention here only several large surveys [13], [10], [3], [17]and two recent representative papers on rational interpolation [7, 18] and on listdecoding of algebraic codes [19].

Many fundamental algorithms for polynomial and rational computations canbe seen as algorithms for matrices with structure. Examples include Toeplitz [ti−j ],Hankel [hi+j−2], Vandermonde [xj−1

i ], and Cauchy matrices [1/(xi− yj)], shown inTable 1.

A SUPERFAST ALGORITHM FOR CONFLUENT TANGENTIAL INTERPOLATION 3

Multiplication of these matrices with vectors often has an analytic interpreta-tion. For instance, the problem of multipoint evaluation of a rational function

a(x) =n∑

k=1

ak

x− yk

at the points x1, . . . , xn is clearly equivalent to computing the product of a Cauchymatrix by a vector:

a(x1)a(x2)

...a(xn)

=

1x1−y1

1x1−y2

· · · 1x1−yn

1x2−y1

1x2−y2

· · · 1x2−yn

......

...1

xn−y1

1xn−y2

· · · 1xn−yn

·

a1

a2

...an

.

Table 2 lists some further analytic interpretations for various matrices with struc-ture relating them to several fundamental algorithms. Its last column lists runningtimes of the corresponding algorithms. Here, we denote by

(1.1) M(n) =

n log n if the field K supportsFFT’s of length n

n log n log log n otherwise

the running time of basic polynomial manipulation algorithms such as multiplica-tion and division with remainder of polynomials of degree < n, cf. [1, Th. (2.8),Th. (2.13), Cor. (2.26)].

The running times in Table 2 are well known. We only mention that theproblem of fast multiplying a Cauchy matrix by a vector is known in the numericalanalysis community as the Trummer problem. It was posed by G. Golub and solvedby Gerasoulis in [6]. We also mention the celebrated fast multipole method (FMM)of [20] that computes the approximate product of a Cauchy matrix by a vector (theFFM method is important in computaional potential theory).

In this extended abstract we continue the work of our colleagues and propose anew superfast algorithm to multiply by a vector a confuent Cauchy-like matrix (afar reaching generalization of a Cauchy matrix). To introduce confluent Cauchy-likematrices we need a concept of displacement recalled next.

1.2. More general matrices, displacement structure. Many applicationsgive rise to the more general classes of structured matrices defined here. We startwith a simple clarifying example. Let us define two auxiliary diagonal matricesAζ = diag{x1, . . . , xn}, and Aπ = diag{y1, . . . , yn}. It is immediate to see that fora Cauchy matrix [1/(xi − yj)], the matrix

AζC − CAπ =[

xi−yj

xi−yj

]=

1 1 · · · 11 1 · · · 1...

......

1 1 · · · 1

is the all-one matrix, and hence rank(AζC − CAπ) = 1.This observation is used to define a more general class of matrices which have

appeared in many applications, e.g., [4, 18] and which have attracted much at-tention recently (see, e.g., [17] and the references therein). For these matrices the


Toeplitz-like matrices R rank (ZR−RZ) << nHankel-like matrices R rank (ZR−RZT ) << nVandermonde-likematrices R rank (D−1

x R−RZT ) << nCauchy-like matrices R rank (DxR−RDy) << n

Table 1. Definitions of basic classes of structured matrices

parameter

(1.2) α = rank(AζC − CAπ),

is larger than 1, but it is still much less than the size of C. Such matrices arereferred to as Cauchy-like matrices.

Similar observations can be made for all other patterns of structure discussedabove. Simply for different kinds of structured matrices we need to use differ-ent auxiliary matrices {Aπ, Aζ}. Table 3 contains definitions for basic classes ofstructured matrices. Here

Z =

0 · · · 0 01 0 · · · 0...

. . . . . ....

0 · · · 1 0

, Dx = diag(x1, · · · , xn).

Matrices in Table 3 are called matrices with displacement structure, the numberα in (1.2) is called the displacement rank. The name “displacement” originatesin signal processing literature [14, 5, 15] where Toeplitz and Hankel matricesare of special interest. For these matrices the auxiliray matrices {Aζ , Aπ} areshift (or displacement matrices) Z. The name displacement structure is now usedalso in connection to other classes of structured matrices in Table 3, though thisterminology is not uniform (For example, in interpolation literature [2] they arecalled null-pole coupling matrices, in [3] they are referred to as matrices with lowscaling rank).

1.3 A generator and superfast multiplication algorithms. It is nowwell-understood [10], [3], [13], [17] that a useful approach to design fast matrixalgorithms is in avoiding operations on n2 entries, and in operating instead on whatis called a generator of a structured matrix.

If the displacement rank (1.2) of a structured matrix R is α, one can factor(non-uniquely)

n︷︸︸︷Aζ R − R Aπ

=−

α︷︸︸︷Bζ

Cπ

(1.3)

where the two rectangular α×n and n×α matrices {Cπ, Bζ} are called a generatorof R.


Toeplitz-like αM(n)Hankel-like αM(n)Vandermonde-like αM(n) log nCauchy-like αM(n) log n

inverses of Toeplitz-like αM(n) log ninverses of Hankel-like αM(n) log n

inverses of Vandermonde-like αM(n) log2 n

inverses of Cauchy-like αM(n) log2 nTable 2. Complexities of multiplication by a vector for matriceswith displacement structure

Let the two auxiliary matrices {Aπ, Aζ} be fixed. If the displacement equation(1.3) has the unique solution R, then the entire information on n2 entries of R isconveniently compressed into only 2αn entries of its generator {Cπ, Bζ}.1

Avoiding operations on matrix entries and operating directly on a generatorallows us to design fast and superfast algorithms. In particular, superfast algorithmsto multiply by vectors matrices in Table 3 can be found in [9]. Table 4 lists thecorresponding complexity bounds.

Notice that the problems of multiplying with the inverse is equivalent to solvingthe corresponding linear system of equations.

1.4. First problem: matrix-vector product for confluent Cauchy-likematrices. Notice that the auxiliary matrices {Aζ , Aπ} in Table 3 are all either shiftor diagonal matrices, i.e., they all are special cases of the Jordan canonical form.Therefore it is natural to consider the more general class of structured matrices Rdefined by using the displacement equation rank(AζR−RAπ) = BζCπ of the form(1.3), where

Aζ = diag{Jm1(x1)⊕ . . .⊕ Jms(xs)}T ,

(1.4) Aπ = diag{Jk1(y1)⊕ . . .⊕ Jkt(yt)},are the general Jordan form matrices. We suggest to call such matrices R confluentCauchy-like matrices. Notice that the class of confluent Cauchy-like matrices isthe most general class of structured matrices, containing all other classes listed inTable 3 as special cases2. Therefore it is of interest to design a uniform superfastalgorithm to multiply a confluent Cauchy-like matrix by a vector: such an algorithmwould then contain all algorithms in Tables 2 and 4 (e.g., convolution, FFT, rationaland polynomial multipoint evaluation and interpolation) as special cases.

Though such a generalized problem is of interest by itself, we were motivatedto study it by a rather general tangential interpolation problem formulated in thenext section.

1Because of page limitations we do not provide details on what happens if there are multiplesolutions R to the displacement equation. We only mention that our algorithm admits a modifica-tion to handle this situation, and that all our complexity bounds fully apply to this more involvedcase. We would like to mention an interesting connection of this case to the rational interpolationproblem discussed in Sec. 1.5 below. Specifically, this degenerate case corresponds to the case ofboundary interpolation treated in [12].

2It also contains new classes of matrices not studied earlier, e.g., scaled confluent Cauchymatrices defined in Sec. 2 below.


1.5. Second problem: Confluent rational matrix interpolation prob-lem. Rational functions appear as transfer functions of linear time-invariant sys-tems, and in the MIMO (Multi-Input Multi-Output) case the corresponding func-tion is a rational matrix function (i.e., an N × M matrix F (z) whose entries arerational functions pij(z)

qij(z) ). It is often important to identify the transfer function F (x)via certain interpolation conditions, one such rather general problem is formulatedbelow. There is an extensive mathematical and electrical engineering literature onsuch problems, some of the pointers can be found in [18].

Tangential confluent rational interpolation problem.

Given:

r distinct points {zk} in the open right-half-plane Π+,with their multiplicities {mk}.

r nonzero chains of N × 1 vectors {xk,1, . . . , xk,mk},

r nonzero chains of M × 1 vectors {yk,1, . . . , yk,mk}.

Construct: a rational N ×M matrix function F (x) such that(1) F (z) is analytic inside the right half plane (i.e., all the poles are in

the left half plane).(2) F (z) is passive, which by definition means that

(1.5) supz∈Π+∪iR

‖F (z)‖ ≤ 1.

(3) F (z) meets the tangential confluent interpolation conditions (k =1, 2, . . . , r):

[xk1 . . . xk,mk

]

F (zk) F ′(zk) · · · F (mk−1)(zk)(mk−1)!

0 F (zk). . .

......

. . . F ′(zk)0 · · · · · · F (zk)

=

(1.6)[

yk,1 · · · yk,mk

].

We next briefly clarify the terminology.

• The passivity condition (1.5) is naturally imposed by the conservationof energy. Indeed, it means that if F (z) is seen as a transfer function ofa certain linear time-invariant system then the energy of the output

y(z) = u(z)F (z)

does not exceed the energy of input u(z).• The term tangential was suggested by Mark Grogorievich Krein. It

means that it is not the full matrix value F (zk) that is given here, ratherthe action of F (zk) for certain directions {xk,1} is prescribed.

• The term confluent emphasizes the condition mk > 1, i.e., not onlyF (z) involved in the interpolation conditions, but also its derivativesF ′(z), . . . , F (mk−1)(z).

We next consider several clarifying special cases to show the fundamental natureof the above interpolation problem.


Example 1.1. The tangential Nevanlinna-Pick problem. In the caseof simple multiplicities mk = 1 the interpolation condition (1.6) reduces to the usualtangential (i.e., x’s and y’s are vectors) interpolation condition

xk · F (xk) = yk

which in the case of the scalar F (z) further reduces to the familiar interpolationcondition of the form

F (xk) =yk

xk.

Example 1.2. The Caratheodory-Fejer problem. Let the vectors{xk,j} are just the following scalars:

[xk,1 · · · xk,mk

]=

[1 0 · · · 0

].

Clearly, in this case (1.6) this is just a Hermite-type rational passive interpolationproblem

F (zk) = yk,1, F ′(zk) = yk,2, · · · F (mk−1)(zk)(mk − 1)!

= yk,mk.

Example 1.3. Linear matrix pencils. Let F (z) = A− zI, then F ′(z) =−I, F ′′(z) = 0. If the condition (1.6) has the form

[uk1 uk2 uk3

]

(A− zkI) −I 00 (A− zkI) −I0 0 (A− zkI)

=

[0 0 0

]

then uk1 is the (left)eigenvector: uk1(A − zkI) = 0, whereas uk2 is the first gen-eralized eigenvector: uk2(A − zkI) = uk1, etc. In this simplest case the confluentinterpolation problem reduces to recovering a matrix from its eigenvalues and eigen-vectors.

To sum up, the confluent tangential rational interpolation problem is a rathergeneral inverse eigenvalue problem that captures several classical interpolationproblems as its special cases.

1.6. Main result. In this paper we describe a superfast algorithm to solvethe confluent tangential rational interpolation problem. The running time of ouralgorithm is

(1.7) Compl(n) = O

(M(n) log n ·

[1 +

r∑

k=1

mk

nlog

n

mk

])

where the multipliciteies {mk} are defined in Sec 1.4, n =∑

mk, and M(n) isdefined in (1.1).

To understand the bound (1.7) it helps to consider two extreme cases.• First, if all mk = 1 (The Nevanlinna-Pick case: n points with simple

multiplicities) then

Compl(n) = M(n) log2(n),

(or n log3(n) if K supports FFT).


• Secondly, if m1 = n (The Caratheodory-Fejer case: one point with fullmultiplicity) then

Compl(n) = M(n) log n,

(or n log2 n if K supports FFT).The algorithm is based on a reduction of the above analytical problem to a

structured linear algebra problem, namely to the problem of multiplication by avector of a confluent Cauchy-like matrix. The corresponding running time is shownto be

O(M(n) · [1 +r∑

k=1

mk

nlog

n

mk+

s∑

k=1

tkn

logn

tk]),

where {mk} are the sizes of the Jordan blocks of Aζ and {tk} are the sizes of theJordan blocks of Aπ, see, e.g., (1.4).

1.7. Derivation of the algorithm and the structure of the paper.The overall algorithm is derived in several steps, using quite different techniques.Interestingly, sometimes purely matrix methods are advantageous, and other timesanalytic arguments help.

(1) First, the confluent tangential rational interpolation problem is reducedto the problem of multiplying by a vector a confluent Cauchy matrix Rwhose generator is composed from the interpolation data. Namely, R isdefined via

AζR + RA∗ζ = BζJB∗ζ ,

where

Aζ =

Jm1(z1)T

Jm2(z2)T

. . .

Jmr (zr)T

,

Bζ =

x11 −y11

......

x1,m1 −y1,m1

......

xr1 −yr1

......

xr,mn −yr,mn

, J =[

IM 00 −IN

]

(2) Secondly, the problem for the confluent Cauchy-like R above is reducedto the analogous problem for the scaled confluent Cauchy matrix (i.e.,not -like) defined in section 2.

(3) Then the problem for the scaled confluent Cauchy matrix is further re-duced in Sec. 3 to the following two problems. One is to multiply aconfluent Vandermonde matrix by a vector, and the second is to mul-tiply the inverse of a confluent Vandermonde matrix by a vector. Thesolution for the second problem is available in the literature (Hermite-type polynomial interpolation).


Ci,j =

B(xi, yj) ∂1yB(xi, yj) ∂2

yB(xi, yj) · · · ∂kj−1y B(xi, yj)

∂1xB(xi, yj) ∂1

x∂1yB(xi, yj) ∂1

x∂2yB(xi, yj) · · · ∂1

x∂kj−1y B(xi, yj)

∂2xB(xi, yj) ∂2

x∂1yB(xi, yj) ∂2

x∂2yB(xi, yj) · · · ∂2

x∂kj−1y B(xi, yj)

.

.....

.

.....

∂mi−1x B(xi, yj) ∂

mi−1x ∂1

yB(xi, yj) ∂mi−1x ∂2

yB(xi, yj) · · · ∂mi−1x ∂

kj−1y B(xi, yj)

,

Figure 3. The structure of Cij in (2.9)

(4) The solution for the remaining problems (multiplication of a confluentVandermonde matrix by a vector) is equivalent to the problem of mul-tipoint Hermite-type evaluation, and the algorithm for it is described inthe last section 4.

2. Scaled confluent Cauchy matrices

2.1. Definition. Suppose we have two sets of n nodes each

{n︷︸︸︷

x1, . . . , x1︸︷︷︸m1

, . . . , xs, . . . , xs︸︷︷︸ms

}

{n︷︸︸︷

y1, . . . , y1︸︷︷︸k1

, . . . , yt, . . . , yt︸︷︷︸kt

},

so that n = m1 +m2 + . . .+ms, and n = k1 +k2 + . . .+kt. We do not assume thatthe s + t nodes x1, . . . , xs, y1, . . . , yt are pairwise distinct. For a scalar bivariatefunction

(2.8) B(x, y) =b(x)− b(y)

x− y,

where

b(x) = (x− y1)k1 · (x− y2)k2 · . . . · (x− ys)kt

we define the block matrix

(2.9) C =[

Ci,j

]1≤i≤s,1≤j≤t

,

where the mi × kj block Ci,j has the form shown in Table 5, where we denote

(2.10) ∂kz :=

1k!

∂k

∂zk.

Before giving a special name to C let us notice that we do not assume that{x1, . . . , xs, y1, . . . , yn} are pairwise distinct, so in the case xi = yj the denominatorin (2.8) is zero, hence we need to clarify what we mean by B(xi, xi) and its partialderivatives. Using the following combinatorial identity

∂px ∂q

yB(z, z) = ∂p+q+1z b(z).


we see that in the case xi = yj the definition (2.8) implies that the block Cij is aHankel matrix of the following form

0 · · · 0 ×... . . . . . .

...

0 . . ....

× ......

...× · · · · · · ×

if mi > kj or

0 · · · · · · · · · 0 ×... . . . . . .

...0 · · · 0 × · · · ×

if mi < kj .For example, if mi > kj then Ci,j has the following Hankel structure:

Cij =

0 · · · 0 ∂kj b(xi)... . . . . . . ∂

kj+1x b(xi)

0 ∂kjx b(xi) . . .

...∂

kjx b(xi) ∂

kj+1x b(xi) · · · ∂

2kj−1x b(xi)

∂kj+1x b(xi)

......

......

...∂mi

x b(xi) ∂mi+1x b(xi) · · · ∂

mi+kj−1x b(xi)

.

We shall refer to C in (2.9) as a scaled confluent Cauchy matrix, and to explainthe name we consider next a simple example.

Example 2.1. Let mi = kj = 1 (so that s = t = n), and {x1, . . . , xn,y1, . . . , yn} are pairwise distinct. Since b(yk) = 0 by (2.8), we have:

C = diag{b(x1), b(x2), . . . , b(xn)} ·

1x1−y1

· · · 1x1−yn

......

1xn−y1

· · · 1xn−yn

.

The above example presents the case of simple multiplicities; in the generalsituation of higher multiplicities {mi} and {kj} we call C in (2.9) a confluentscaled Cauchy matrix.

2.2. Analytic interpretation. We have mentioned that multiplication ofstructured matrices by a vector often has an analytic interpretation, see, e.g., Ta-ble 2. Confluent Cauchy matrices are not an exception, and multiplying C in (2.9)by a vector

[a0,1 . . . ak1−1,1 . . . a0,t . . . akt−1,t

]T


AζC − CAπ =

b(x1) 0 · · · 0 b(x1) 0 · · · 0∂1

xb(x1) 0 · · · 0 · · · · · · ∂1xb(x1) 0 · · · 0

......

......

......

∂m1−1x b(x1) 0 · · · 0 ∂m1−1

x b(x1) 0 · · · 0

......

......

b(xs) 0 · · · 0 b(xs) 0 · · · 0∂1

xb(xs) 0 · · · 0 · · · · · · ∂1xb(xs) 0 · · · 0

......

......

......

∂ms−1x b(xs) 0 · · · 0 ∂ms−1

x b(xs) 0 · · · 0

Figure 4. Displacement of a scaled confluent Cauchy matrix C

is equivalent to multipoint evaluation of a rational function (with fixed poles {yj}of the orders {tj})

r(x) =t∑

j=1

kj−1∑

l=0

al,j∂lyB(x, yj)

at points {xi} as well as of its derivatives up to the orders {mi}.2.3. Displacement structure. Let the auxiliary matrices {Aζ , Aπ} be de-

fined as in (1.4), and define the class of confluent Cauchy-like matrices as thosehaving a low displacement rank (1.2) with these {Aζ , Aπ}. The following examplejustifies the latter name.

Example 2.2. Our aim here is to show that for the scaled confluent Cauchymatrix R in (2.9) we have

(2.11) rank(AζR−RAπ) = 1

where {Aζ , Aπ} are as in (1.4).Indeed, applying ∂p

x∂qy (recursively for p, q = 0, 1, 2, . . .) to the both sides of

xB(x, y)− yB(x, y) = b(x)− b(y)

we obtainx∂p

x∂qyR(x, y) + ∂p−1

x ∂qyR(x, y)− y∂p

x∂qyR(x, y)−

−∂px∂q−1

y R(x, y) =

0 if p > 0 and q > 0bi(x) if q = 0bj(y) if p = 0

Now arranging the latter equation in a matrix form we obtain

Jmi(xi)T Cij − CijJkj (yj) =

b(xi) 0 · · · 0∂1

xb(xi) 0 · · · 0...

......

∂mi−1x b(xi) 0 · · · 0


Combining such equations for each block Rij together we finally obtain the formulashown in Table 6, which yields (2.11).

Thus, the displacement rank of scaled confluent Cauchy matrices is 1, so wecoin the name confluent Cauchy-like with the matrices whose displacement rank,though higher than 1, still much smaller than the size n.

3. Factorization of confluentCauchy matrices. Confluent

Vandermonde matrices

We shall need a superfast algorithm to multiply a confluent Cauchy matrix bya vector, this algorithm will be based on the formula (3.12) derived next.

Theorem 3.1. Let C be the confluent Cauchy-like matrix defined in (2.9),b(x) is defined in (2.8) and ∂x is defined in (2.10). Then

(3.12) C = VP (x)VP (y)−1diag{B1, B2, . . . Bn},where

VP (x) = col

VP (x1)...

VP (xk)

with VP (xi) =

P0(xi) P1(xi) · · · Pn−1(xi)∂xP0(xi) ∂xP1) · · · ∂xPn−1(xi)∂2

xP0(xi) ∂2xP1(xi) · · · ∂2

xPn−1(xi)...

......

∂mi−1x P0(xi) ∂mi−1

x P1(xi) · · · ∂mi−1x Pn−1(xi)

,

and

(3.13) Bi =

0 · · · 0 ∂m1x b(xi)

... . . . . . . ∂m1x b(xi)

0 ∂m1x b(xi) . . .

...∂m1

x b(xi) ∂(m1+1)x b(xi) · · · ∂

(2m1−1)x b(xi)

,

The proof is based on the following formula which is of interest by itself:

VP (y)−1 = IV TP

(y) · diag (B−11 , B−1

2 , . . . , B−1k ),

where I stands for the flip permutation matrix, and {P} denotes the associatedsystem of polynomials defined in [11] (see also [16] for a connection with inversionof discrete transmission lines) .

We now turn to the computational aspects of the formula (3.12). Though theformula is given for an arbitrary polynomial system {P} we shall use it here only forthe usual power basis Pk(x) = xk (other choices, e.g., the Chebyshev polynomialsare useful from the numerical point of view). The formula reduces the problemof multiplication of C by a vector to the same problem for the three matrices onthe right hand side of (3.12). Each of the three problems is treated next (it isconvenient to discuss these three problem in a reversed order, i.e., first step 3, thenstep 2, and finally step 1).


Step 3. Multipoint Hermite-type evaluation: Theproblem of multiplying V (x) by a vector [ak] is equivalent to the problemof multipoint Hermite-type evaluation. Indeed, let us use the entries [ak]of to define the polynomial a(x) =

∑n−1k=0 akxk. Then it is clear that we

need to evaluate the value of a(x) at each xi as well as the values of itsfirst mi (scaled) derivatives a(s)(xi)

s! (notice that the rows of V (xi) areobtained by differentiation of their predecessors). The algorithm for thisproblem is offered in the next section.

Step 2. Hermite interpolation: The problem of multiplying VP (x)−1

by a vector is the problem that is inverse to the one discussed in thestep 3 above. The superfast algorithm for this problem can be found in[1]. The complexity is the same as in the step 3.

Step 1. Convolution: The problem of multiplication of diag{B1, B2, . . . ,Bn} by a vector is clearly reduced to the convolution, since all diagonalblocks (3.13) have a Hankel structure (constant along anti-diagonals).We also need to compute the entries of all these blocks Bk, i.e., for eachpoint xk we need to compute the first 2mi Taylor coefficients ∂s

xb(xk)(s = 0, 1, . . . 2mi − 1). This is exactly the problem we treat in the step3 above.

4. Multipoint Hermite-type evaluation

4.1. The problem. The problem we discuss here is that of computing theTaylor expansion of a univariate polynomial at different points. More precisely, forf ∈ K[x] and ξ ∈ K there are uniquely determined elements f0,ξ, f1,ξ, . . . such that

f = f0,ξ + f1,ξ(x− ξ) + f2,ξ(x− ξ)2 + · · · .

The sequence T (f, ξ, n) := (f0,ξ, . . . , fn−1,ξ) is called the sequence of Taylor coeffi-cients of f up to order n. Let ξ1, . . . , ξt be elements in K and d1, . . . , dt be positiveintegers, and suppose that f is a univariate polynomial of degree < n := d1+· · ·+dt.We want to develop a fast algorithm for computing the sequences

T (f, ξ, d1), . . . , T (f, ξ, dt).

Our algorithm will use as a subroutine a fast multiple evaluation algorithm. Itsrunning time involves the entropy of the sequence (d1, . . . , dt) defined by

H(d1, . . . , dt) := −t∑

i=1

di

nlog

di

n,

where n =∑

i di and log is the logarithm to the base 2.4.2. Our Algorithm. The algorithm we present for the problem stated above

consists of several steps.

Algorithm 4.1. On input ξ ∈ K and d ≥ 1 the algorithm computes thepolynomials Π`,ξ := (x − ξ)2

`

for ` = 0, . . . , dlog de, as well as the polynomial(x− ξ)d.

(1) Put Π0,ξ := (x− ξ).(2) For ` := 1, . . . , dlog die put Π`,ξ := Π2

`−1,ξ.(3) Let d = 2`1 + 2`2 + . . . + 2`s with `1 < · · · < `s. Put P := Π`1,ξ.(4) For i = 2, . . . , s set P := P ·Π`i,ξ. The final value of P equals (x− ξ)d.


Lemma 4.2. Algorithm 4.1 correctly computes its output with O(M(d))operations.

The proof of this and the following assertions are omitted and will be includedin the final version of the paper.

The next step of our presentation is the solution to our problem in case t = 1.

Algorithm 4.3. On input a positive integer d, and element ξ ∈ K, anda univariate polynomial f ∈ K[x] of degree less than d, the algorithm computesT (f, ξ, d). We assume that we have precomputed the polynomials (x − ξ), (x −ξ)2, . . . , (x− ξ)2

τ

where τ = dlog de.(1) If d = 1, return f , else perform Step (2).(2) Compute f0 := f mod (x − ξ)2

τ−1and f1 := (f − f0)/(x − ξ)2

τ−1and

run the algorithm recursively on f0 and f1. Output the concatenation oftheir outputs.

Lemma 4.4. The above algorithm correcly computes its output in timeO(M(d)).

Now we are ready for our final algorithm.

Algorithm 4.5. On input positive integers d1, . . . , dt, elements ξ1, . . . , ξt ∈K, and a polynomial f ∈ K[x] of degree less than n :=

∑ti=1 di, the algorithm com-

putes the sequences T (f, ξi, di), i = 1, . . . , t.(1) We use Algorithm 4.1 to the inputs (ξ, di) for i = 1, . . . , t. At this stage,

we have in particular computed (x− ξi)di for i = 1, . . . , t.(2) Now we use the multiple evaluation algorithm given in the proof of [1,

Th. (3.19)] to compute fi := f mod (x− ξi)di , i = 1, . . . , t.(3) Use Algorithm 4.3 on input (f, ξi, di) to compute T (f, ξi, di) for i =

1, . . . , t.

Theorem 4.6. The above algorithm correctly computes its output in timeO(M(n)H) where H is the entropy of the sequence (d1, . . . , dt).

We remark that the algorithm can obviously be customized to run in paralleltime O(H) on O(n) processors.

References

[1] P. Burgisser, M. Clausen and A. Shokrollahi, Algebraic Complexity Theory, series =Grundlehren der Mathematischen Wissenschaften, vol. 315, Springer Verlag, Heidelberg,1996.

[2] J. Ball, I. Gohberg and L. Rodman, Interpolation of rational matrix functions, OT45,Birkhauser Verlag, Basel, 1990.

[3] D. Bini and V. Pan, Polynomial and Matrix Computations, Volume 1, Birkhauser, Boston,1994.

[4] Ph. Delsarte, Y. Genin and Y. Kamp, On the role of the Nevanlinna-Pick problem in circuitand system theory, Circuit Theory and Appl., 9 (1981), 177-187.

[5] B. Friedlander, M. Morf, T. Kailath and L. Ljung, “New inversion formulas for matricesclassified in terms of their distance from Toeplitz matrices,” Linear Algebra and Appl., 27,31-60, 1979.

[6] A. Gerasoulis, A fast algorithm for the multiplication of generalized Hilbert matrices withvectors, Math. of Computation, 50 (No. 181), 1987, 179 – 188.

[7] I. Gohberg and V. Olshevsky, “Fast state space algorithms for matrix Nehari and Nehari-Takagi interpolation problems,” Integral Equations and Operator Theory, 20, No. 1, pp. 44-83, 1994.


[8] I. Gohberg and V. Olshevsky, Fast inversion of Chebyshev-Vandermonde matrices, Nu-merische Mathematik, 67 (1) (1994), 71-92.

[9] I. Gohberg and V. Olshevsky, Complexity of multiplication with vectors for structured ma-trices, Linear Algebra and Its Applications, 202 (1994), 163-192.

[10] G. Heinig and K. Rost, Algebraic methods for Toeplitz-like matrices and operators, OperatorTheory, vol 13, Birkhauser, Basel, 1984.

[11] T. Kailath and V. Olshevsky, Displacement Structure Approach to Polynomial Vandermondeand Related Matrices, Linear Algebra and Its Applications, 261 (1997), 49-90.

[12] T. Kailath and V. Olshevsky, Diagonal Pivoting for Partially Reconstructible Cauchy-likeMatrices, With Applications to Toeplitz-like Linear Equations and to Boundary RationalMatrix Interpolation Problems, Linear Algebra and Its Applications, 254 (1997), 251-302.

[13] T. Kailath and A.H. Sayed, Displacement structure : Theory and Applications, SIAM Review,37 No.3 (1995), 297-386.

[14] M.Morf, “Fast algorithms for multivariable systems,” Ph.D. thesis, Department of ElectricalEngineering, Stanford University, 1974.

[15] M.Morf, “Doubling Algorithms for Toeplitz and Related Equations,” Proc. IEEE Internat.Conf. on ASSP, pp. 954-959, IEEE Computer Society Press, 1980.

[16] V. Olshevsky, Eigenvector computation for almost unitary Hessenberg matrices and inver-sion of Szego-Vandermonde matrices via discrete transmission lines. Linear Algebra and ItsApplications, (285)1-3 (1998) pp. 37-67

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[18] V. Olshevsky and V. Pan, “A superfast state-space algorithm for tangential Nevanlinna-Pick interpolation problem,” in Proceedings of the 39th IEEE Symposium on Foundations ofComputer Science, pp. 192–201, 1998.

[19] V. Olshevsky and A. Shokrollahi, “A Displacement Approach to Efficient Decoding of Alge-braic Codes,” in Proceedings of the 31st Annual ACM Symposium on Theory of Computing,235–244, 1999.

[20] V. Rokhlin, Rapid Solution of Integral Equations of Classical Potential Theory, J. Com-put. Physics, 60, 187–207, 1985.

Department of Mathematics and Statistics, Georgia State University, Atlanta, GA30303

E-mail address: [email protected]

Digital Fountain, 600 Alabama Street, San Francisco, CA 94110E-mail address: [email protected]

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