From: Ned Anderson <firstname.lastname@example.org>
Date: Mon, 22 Mar 93 07:25:11 PST
Subject: The L(D**(-1),U) Decomposition.
High Performance Computing: Using an L(D**(-1),U) decomposition.
This note proposes the use of a modified approach to information storage
in the LU decomposition process for solving linear systems. This approach
has significant practical value, particularly in the solution of
tridiagonal or other banded systems.
Simply stated, the proposal is to, at each elimination step:
1. Use the reciprocal of the current pivot element to do the elimination
(this is already a common technique).
2. Store this reciprocal (of the diagonal element of U) in place of
the diagonal element itself.
I propose we call this approach the L(D**(-1),U) decomposition.
This lets us replace the N floating divides in the back solve by N multiplies.
Its value derives from the fact that on many machines, divides are much
slower than multiplies, and/or are not part of the pipelined arithmetic unit.
Here is a code segment (cf. xGTSV in LAPACK) implementing the proposed
approach for a tridiagonal solver (with no pivoting or storage of L):
D( 1 ) = ONE / D( 1 )
DO K = 1, N - 1
MULT = DL( K ) * D( K )
D( K+1 ) = D( K+1 ) - MULT*DU( K )
B( K+1 ) = B( K+1 ) - MULT*B( K )
D( K+1 ) = ONE / D( K+1 )
* D**(-1),U and (L**(-1))B calculations complete.
* Back solve:
B( N ) = B( N ) * D( N )
DO K = N - 1, 1, -1
B( K ) = ( B( K )-DU( K )*B( K+1 ) ) * D( K )
This code yielded a 50% improvement in the Mflop rate (over the traditional
algorithm) on a DEC Alpha AXP 150 Mhz machine.
For full matrices and LAPACK, one may want to consider modifying xGETF2
to store reciprocal pivots, and modifying the BLAS3 routine xTRSM to accept
a third value, 'reciprocal' for the DIAG parameter which now accepts either
'unit' or 'nonunit'.
-- Dr. Ned Anderson, Digital Equipment Corporation, (508)467-4862 3/3/93
[Editor's Note: It should be mentioned that this approach is slightly
less accurate and robust. Consider the 1-by-1 case. Solve 7*x = 21.
Computing x = 21/7 = 3 gives the exact answer, while computing
x = 7**(-1)*21 = (.142857...)*21 might not give an exact 3. Moreover,
some pivots, near the under- or overflow limits, cannot be reciprocated.
But, neither is these points is actually very important. If you want speed,
From: Grace Wahba <email@example.com>
Date: Mon, 22 Mar 93 12:02:37 -0600
Subject: Reply about Fitting Experimental Data
In response to C. Zhu concerning fitting experimental
data by best approximation, see my book `Spline Models for
Observational Data' SIAM 1990 and 13 pages of references cited
there. (Order info may be found by ftp at ftp.stat.wisc.edu in
pub/wahba/bkinfo). He might also want to look at the literature
on minimax estimation in the statistical literature.
From: John Prentice <firstname.lastname@example.org>
Date: Wed, 24 Mar 93 17:41:37 MST
Subject: Snow Avalanche Info Needed
This question is more a physics question than a computing
or mathematics question, but since this newsletter goes to
a very diverse group of people, I thought I would submit
it anyway and see if anyone has any thoughts.
I am doing some research into the solid/fluid mechanics of
snow avalanches, particularly avalanches in the continental
region of the US Rockies. To carry out my research program,
I need to find out just about everything that is known about
the constitutive properties and physics of snow, particularly
highly porous snow such as we get in this region. I also need to
find out what is known about the constitutive properties and
physics of hoar depth and phase transitions in snow in general.
If anyone has any references, pointers, or recommendations for
who to ask, I would much appreciate it. I am also trying to
dig up any books or articles on the general physics of snow
avalanches. Any recommendations for that would also be much
Dr. John K. Prentice
Quetzal Computational Associates
3200 Carlisle N.E., Albuquerque, NM 87110-1664 USA
Phone: 505-889-4543 E-mail: email@example.com -or- firstname.lastname@example.org
From: Stefano Foresti <email@example.com>
Date: Thu, 25 Mar 1993 15:00:28 -0700
Subject: Finite Element Codes with Message Passing
We are interested in public domain finite element codes that have been
developed for distributed memory parallel computers with message
passing software (for instance Intel, PVM, Parasoft Express, etc.).
Any information is appreciated.
Stefano Foresti and Dave Barker
Utah Supercomputing Institute
University of Utah
Salt Lake City, Utah 84112, USA
tel: (801)581-3173 Fax: (801)585-5366
From: Bill Mitchell <firstname.lastname@example.org>
Date: Fri, 26 Mar 93 10:49:54 EST
Subject: MGGHAT Now Available
MGGHAT (MultiGrid Galerkin Hierarchical Adaptive Triangles) is a program
for the solution of linear second order elliptic partial differential equations
of the form:
-( p * u ) -( q * u ) + r * u = f in omega
x x y y
u = g on boundary 1 of omega
u + c * u = g on boundary 2 of omega
where p,q,r,f,c and g are functions of x and y, omega is a polygonal
domain in R2, n is the outward unit normal, and boundaries 1 and 2
form a partition of the boundary of omega.
This program is now available for public use. It uses a finite element method
with linear, quadratic or cubic elements over triangles. The adaptive
refinement via newest vertex bisection and the multigrid iteration are both
based on a hierarchical basis formulation. Visualization is available at run
time through an X Windows display, and a posteriori through output files that
can be used as GNUPLOT input.
The user supplies subroutines containing the coefficient functions p, q, r,
f, c, and g, a subroutine with the inital (coarse grid) triangulation to
define the domain, and a main program that sets program parameters and calls
subroutine mgghat. The program is mostly FORTRAN 77, but uses C for the
X11 graphics (can be disabled).
MGGHAT is available from netlib. Send the message
"send index from pdes/mgghat" to email@example.com.
MGGHAT can also be obtained from MGNET (multigrid network) by annonymous
ftp to casper.cs.yale.edu in directory mgnet/mgghat.
William F. Mitchell | firstname.lastname@example.org
GE Advanced Technology Labs | email@example.com
Bldg. 145-2 Moorestown Corp. Center | Voice: (609) 866-6536
Moorestown, NJ 08057 | Fax: (609) 866-6397
From: Volker Mehrmann <firstname.lastname@example.org>
Date: Tue, 23 Mar 93 14:15:42 MET
Subject: Change of Address for Volker Mehrmann
On April 1, 1993 I will move to Chemnitz and my new address will be
Technische Universitaet Chemnitz-Zwickau
Reichenhainer Str. 41
Tel. (049) (0)371-561-2659 (office)
(049) (0)371-561-2657 (fax)
From: Edoh Y. Amiran <EDOH@NESSIE.CC.WWU.EDU>
Date: 26 Mar 1993 14:02:07 -0800 (PST)
Subject: Pacific Northwest Dynamical Systems Seminar
The Pacific Northwest Dynamical Systems Seminar will be held at Western
Washington University in Bellingham, WA, on April 17-th. The speakers are:
M. Gerber (U. IN), F. Diacu (U. Vic.), D. Chalice (WWU), S. Tuncel (UW),
E.S. VanVleck (SFU), and K. Lu (BYU).
The talks will be held at Parks Hall 164 from 9:00 to 5:00.
For more information e-mail email@example.com or write to Edoh Amiran, Mathematics,
WWU, Bellingham WA, 98225-9063.
-- Edoh Y. Amiran
From: Gene Golub <golub@sccm.Stanford.EDU>
Date: Sat, 27 Mar 93 17:25:51 PST
Subject: Bay Area NA Day
ANNUAL BAY AREA NA DAY
The annual Bay Area NA Day will take place at Stanford on Sat, April
10, beginning at 9:30 am. The meeting will take place in History
Corner. The speakers will include Alan Edelman, Vince Fernando,
John Strain, Andrew Stuart and several graduate students from Berkeley
There will be a reception at Gene Golub's house followed by a Chinese
banquet; the cost is $20 per person. All interested persons are
welcome. Please send me a note by April 7 if you are interested in
attending, and participating in the dinner.
For further information, call me at (415) 723-3124 or send an e-mail msg.
From: Nancy K. Nichols <SMSNICHO@cms.am.cc.reading.ac.uk>
Date: Wednesday 24th March 1993 18:17:31 GMT
Subject: Symposium in Honour of Leslie Fox
A SYMPOSIUM IN HONOUR OF LESLIE FOX
together with the
SIXTH LESLIE FOX PRIZE COMPETITION
is to be held
JUNE 24 - 25, 1993
Mathemtatical Institute, Oxford University
The symposium is being held to celebrate the many contributions of Leslie
Fox to the research, teaching and applications of numerical analysis.
The programme includes: on Thursday, June 24, the finals of the Sixth
Leslie Fox Prize Competition and the award of the prizes by Clemency Fox.
An invited lecture by Professor G.H. Golub is to follow. On Friday,
June 25, invited talks on topics in numerical analysis are to be
presented by friends and colleagues associated with different stages of
Leslie's career, from schooldays through retirement. Speakers include:
Dr. D. Kershaw, Professor C. Clenshaw, Professor J. Walsh, Dr. A. Tayler,
Dr. D. Mayers, Dr. N.I.M. Gould, and Professor A. Stuart.
A sherry reception and honorary dinner are to be held in Balliol College
on the evening of Thursday, June 24. All friends are welcome at the dinner
(including those not participating in the scientific programme).
Registration fees for the conference are:
Two day programme: #40 including lunches in Balliol College, morning
and afternoon coffee/tea and sherry reception (#20 for students)
Fox Prize day only: #20 including lunch, coffee/tea and reception
(#10 for students)
Conference Dinner: #26 (including wine) for participants
#30 (including reception and wine) for non-participants
Accomodation for a limited number of participants is to be available in
Balliol College at a rate of #19.40 per person for bed and breakfast.
Information on other accomodation can be obtained from the symposium
Any funds raised from the meeting are to be contributed to
the Leslie Fox Prize Fund. Additional donations to the capital fund
are welcome at any time. Cheques should be made payable to the
Secretary, IMA, 16 Nelson Street, Southend-on-Sea, Essex SS1 1EF, UK.
Local Organizers: Prof. K.W. Morton (Chairman), Dr. N.K. Nichols,
Dr. D. Mayers, Prof. S. Hammarling, Prof. I.S. Duff
Fox Prize Adjudicators: Dr. N.K. Nichols (Chairman), Prof. C.M. Elliott,
Prof. C.T.H. Baker
Those planning to attend the symposium are encouraged to register
as soon as possible.
Correspondence, including a request for an electronic registration form,
should be addressed to either:
Professor K.W. Morton Dr. N.K. Nichols
Oxford University Department of Mathematics
Computing Laboratory University of Reading
8/11 Keble Road Box 220
Oxford OX1 3DQ UK Reading RG6 2AX UK
Tel. +44 865 273885 Tel. +44 734 318988
FAX +44 865 273839 FAX +44 734 313423
From: A. Spence <A.Spence@maths.bath.ac.uk>
Date: Fri, 19 Mar 93 15:31:41 GMT
Subject: Numerical Analysis and Dynamical Systems
LONDON MATHEMATICAL SOCIETY
TWO DAY MEETING
THURSDAY -- FRIDAY 13--14 MAY 1993
UNIVERSITY OF CAMBRIDGE
Numerical Analysis and Dynamical Systems
Thursday 2.00 Opening Session
2.10 C. Foias (Bloomington, Indiana): ``Interpolation
theory and the localization of the global attractor''
3.45 C. Grebogi (College Park, Maryland): ``Shadowing in
5.00 D. Watkins (Seatle, Washington): ``A numerical
analyst's view of isospectral flows''
Friday 10.15 R.D. Skeel (Urbana-Champaign, Illinois):
``Symplectic integration with multiple time steps
and variable step size''
12.00 C.M. Elliott (Sussex): ``Dynamical phase transitions
and parabolic equations''
2.30 E. Doedel (Caltech): ``On algorithms and software for
3.30 Meeting closes
The lectures will be held at the Mill Lane Lecture Rooms, (opposite
DPMMS) Cambridge. All interested are very welcome.
A dinner will be held at The University Centre on the evening of Thursday
13th May at 8.00pm. The cost of the dinner will be 18.50 pounds per person,
inclusive of wine. Those wishing to attend should inform A.
Iserles, DAMTP, Silver Street, Cambridge CB3 9EW, enclosing a cheque payable
to ``University of Cambridge'', to arrive by 7th May.
Enquiries may be addressed to A. Iserles, DAMTP, Silver Street, Cambridge,
CB3 9EW (email A.Iserles@uk.ac.cam.amtp) or A. Spence, School of Mathematical
Sciences, University of Bath, Claverton Down, Bath, BA2 7AY (email
From: Annie Cuyt <firstname.lastname@example.org>
Date: Thu, 25 Mar 93 14:53:15 +0100
Subject: Nonlinear Numerical Methods and Rational Approximation
Nonlinear Numerical Methods and Rational Approximation
Organiser: A. Cuyt (UIA, Antwerp).
Scientific Committee: A. Bultheel (KUL, Leuven), A. Cuyt (UIA, Antwerp),
J. Meinguet (UCL, Louvain-la-Neuve), J.-P. Thiran (FUN, Namur).
UPON GENERAL REQUEST THE DEADLINE FOR REGISTRATION IS EXTENDED TO
MAY 1. SINCE I AM ALSO ABROAD IN APRIL, THERE WILL BE NO REPLY TO
ANY REQUESTS FOR INFORMATION IN THAT PERIOD.
Invited Speakers: A. Gonchar (Moscow), M. Gutknecht (Zurich),
W. B. Jones (Boulder, USA), D. Lubinsky (Witwatersrand, S-Africa),
E. Saff (Tampa, USA).
In 1993 another conference will be organised on the use of rational functions
in different fields of numerical analysis. It will take place at the
University of Antwerp (UIA) from September 5 to 11. The registration fee of
7 500 Bf will cover participation, lunch and dinner on campus, all social
events, coffee-breaks, transportation from and to town and a copy of the
conference proceedings. Since Antwerp is in 1993 Europe's cultural capital,
special attention will be paid to housing and social events for the
participants. Registration should be made before April 1 on the enclosed form.
Each of the following sessions will be introduced by a one-hour survey
lecture. All other participants are invited to present a 20-minute
research talk. Abstracts should be mailed together with the registration
form before April 1. Proceedings will be published by Reidel just like in
1987. Only papers received in their final form by September 30 can be
considered for publication. Details for the preparation of a manuscript
shall be mailed to all registered participants after June 1.
Sessions: The emphasis will be on Pade approximation,
Rational interpolation, Rational approximation, Continued fractions
and Orthogonal polynomials. For each of those topics we also welcome
all Multivariate and Multidimensional problems, Applications, Error
analysis and Software development.
Further Information: Annie Cuyt, Dept of Mathematics and
Computer Science, University of Antwerp (UIA), Universiteitsplein 1,
B-2610 Wilrijk-Antwerp, Belgium, tel: (32) 3/820.24.07, fax: (32)
3/820.24.21, secr: (32)3/820.24.01, e-mail: email@example.com
From: Karen Friedman <firstname.lastname@example.org.EDU>
Date: Tue, 23 Mar 93 16:26:06 MST
Subject: National Information Infrastructure
TELECOMMUNICATIONS AS PART OF THE
NATIONAL INFORMATION INFRASTRUCTURE
You are cordially invited to attend the IEEE-USA Information
Exchange on Telecommunications as Part of the National Informa-
The United States needs to develop a world-class national infor-
mation infrastructure. Fundamental to achieving this goal is
establishing a national telecommunications policy that fosters
effective application of evolving technologies, incentives for
investment, and responsiveness to users. Many groups and associ-
ations are initiating projects to represent the views of their
members on such an infrastructure.
Because IEEE's wide-ranging membership touches almost all enti-
ties addressing this issue, the IEEE Committee on Communications
& Information Policy will convene an
May 4-5, 1993
Rosslyn Westpark Hotel
The purpose of this exchange is to allow the groups and associa-
tions developing a vision of U.S. information infrastructure to
share views and better understand their respective issues. We
envision a dialogue among those already active in this arena and
other key individuals who are directly concerned with telecommun-
ications. This will be a neutral ground where different
interests can target particular issues important to them and this
country. Panelists of national stature will cover the range from
technology through policy. We expect many officials of the Exe-
cutive and Legislative branches to participate as well.
The ultimate goal of this effort is to bring together the ele-
ments for a national policy to guide the nature and continuing
development of a superior infrastructure. This Information
Exchange is a first step in the process.
Attendance is limited. The conference fee of $150.00 and the
completed registration form are due by April 15. A block of
rooms has been reserved in the name of "IEEE" at a discounted
rate of $87.00. Hotel reservations must be made directly with
the hotel at [PHONE (703) 527-4814; FAX (703) 522-8864] or before
April 3, 1993.
For registration and housing information, and a copy of the
advance program, please contact:
Manager, Technology Policy Council
1828 L Street NW, Suite 1202
Washington, DC 20036
PHONE: (202) 785-0017
FAX: (202) 785-0835
From: Xian-He Sun <email@example.com>
Date: Mon, 22 Mar 93 21:34:54 -0500
Subject: Final Call for Papers: JPDC special issue
CALL FOR PAPERS
Special Issue of the Journal of Parallel and Distributed Computing on
Scalability of Parallel Algorithms and Architectures.
Papers are solicited for a special issue of the Journal of Parallel and
Distributed Computing on scalability of parallel algorithms and
architectures planned for April 1994. The issue will publish about
eight regular papers (about 35 double-spaced manuscript pages) and
upto four research notes (about 15 double-spaced manuscript pages).
The objective of this special issue is to bring together the research
done by theoreticians, application developers, and designers of
architectures and system software on scalability analysis of large-
scale parallel systems. Topics of interest include but are not limited
to Scalability metrics, Scalability analysis of parallel algorithms,
applications, and architectures, Cost effectiveness of scaling parallel
architectures in terms of performance, Software tools for scalable
Authors should follow the JPDC manuscript format as described at the
end of each JPDC issue. Five copies of complete manuscript should be
sent to one of the guest editors by April 5, 1993. The authors will be
notified of the decision by September 1, 1993 and the final manuscript
ready for publication will be due by October 1, 1993.
Prof. Vipin Kumar Dr. Xian-He Sun
Department of Computer Science ICASE
200 Union Street S.E., Rm. 4-192 Mail Stop 132C
University of Minnesota NASA Langley Research Center
Minneapolis MN 55455 Hampton, VA 23681-0001
Tel: (612) 624-8023 Tel: (804) 864-8018
Fax: (612) 625-0572 Fax: (804) 864-6134
Email: firstname.lastname@example.org email@example.com
From: Daniel Baltzer <firstname.lastname@example.org>
Date: Mon, 22 Mar 1993 16:07:21 -0700
Subject: Contents, Numerical Algorithms
Contents NUMERICAL ALGORITHMS
Editor-in-Chief: Claude Brezinski, Laboratoire d'Analyse Numerique et
d'Optimisation, UFR IEEA - M3, Universite de Lille 1, France, fax: +33 - 20
43 49 95, e-mail: email@example.com
Numerical Algorithms, Volume 3 (1993), issues 1-4:
Moment problem formulation of the Schrodinger equation, D. Bessis and C. R.
Extrapolation methods in numerical integration, P. Rabinowitz
A reliable modification of the cross rule for rational Hermite
interpolation, B. Beckermann and C. Carstensen.
A uniform approach for Hermite Pade and simultaneous Pade approximants and
their matrix-type generalizations, B. Beckermann and G. Labahn.
Interpolatory integration rules and orthogonal polynomials with varying
weights, T. Bloom, D.S. Lubinsky and H. Stahl.
Para-orthogonal Laurent polynomials and associated sequences of rational
functions, C.M. Bonan-Hamada, W.B. Jones, W.J. Thron and A. Magnus.
Some vector sequence transformations with applications to system of
equations, C. Brezinski and H. Sadok.
A Favard theorem for orthogonal rational functions on the unit circle, A.
Bultheel, P. Gonzalez-Vera, E. Hendriksen and O. Njastad.
A moment problem associated to rational Szego functions, A. Bultheel, P.
Gonzalez-Vera, E. Hendriksen and O. Njastad.
Orthogonal rational functions and quadrature on the unit circle, A.
Bultheel, P. Gonzalez-Vera, E. Hendriksen and O. Njastad.
Rational approximants to symmetric formal Laurent series, M. Camacho and P.
Rational interpolation with a single variable pole, J. M. Carnicer.
The Neville-Aitken formula for rational interpolants with prescribed poles,
C. Carstensen and G. Muhlbach.
Rational approximations for the cosine function: P-acceptability and order,
Multivariate rational data fitting: general data structure, maximal
accuracy and object orientation, A. Cuyt and B. Verdonk.
Implementation of the recurrence relations of biorthogonality, Z. Da Rocha.
Convergence of simultaneous Hermite-Pade approximants to the n-tuple of
q-hypergeometric series, M. G. de Bruin, K.A. Driver and D.S. Lubinsky.
Bivariate Hermite-Birkhoff interpolation and Vandermonde determinants, M
Gasca and J.J. Martinez.
Matrix orthogonal Laurent polynomials and two-point Pade approximants, C.
Gonzalez-Concepcion, P. Gonzalez-Vera, and E. Hendriksen.
On extrapolation of Jagerman and Stetter rules, S. Gonzalez-Pinto, P.
Gonzalez-Vera and J.C. Santos.
Solution of integral equations using function-valued Pade approximants II,
P.R. Graves-Morris and R. Thukral
Rational approximation to Neumann series of Bessel functions, N. Hayek, P.
Gonzalez-Vera and F. Perez-Acosta.
A Levin-type algorithm for accelerating the convergence of Fourier series,
Asymptotics for Szego polynomial zeros, W.B. Jones, O. Njastad and H. Waadeland.
Evaluation of a rational function, A.G. Law, C.N. Zhang, A. Rezazadeh and
The quadratic convergence of the topological epsilon algorithm for systems
of nonlinear equations, H. Le Ferrand.
Quadratic decomposition of orthogonal polynomials: a matrix approach, F.
Marcellan and G. Sansgire.
Two-dimensional orthogonal polynomials, their associated sets and the
co-recursive sets, P. Maroni.
Convergence and acceleration properties for the vector e-algorithm, A.C. Matos.
On the internal stepsize of an extrapolation algorithm for IVP in ODE,
M.Murofushi and H. Nagasaka.
Extrapolation methods for some singular fixed point sequences, N. Osada.
On higher order Pade-type approximants with some prescribed coefficients in
the numerator, M.A. Pinar and T.E. Perez.
Particular rules for the O-algorithm, M. Redivo Zaglia.
Clifford algebras and vector-valued rational forms II, D.E. Roberts.
P.C.-fractions and Szego polynomials associated with starlike univalent
functions, F. Ronning.
Sobolev inner products and orthogonal polynomials of Sobolev type, A, Ronveaux.
Asymptotic behaviour of iterated modified transforms on some slowly
convergent sequences, P. Sablonniere.
Numerical factorization of a polynomial by rational Hermite interpolation,
T.Sakurai, H. Sugiura and T. Torii.
On extending backwards positive definite sequences, F.H. Szafraniec.
Global extrapolation with a parallel splitting method, X.-C. Tai.
Limit period Schur algorithms, W.J. Thron.
A general module theoretic framework for vector M-Pade and matrix rational
interpolation, M. van Barel and A. Bultheel.
Best choice of the pole for the Pade-type approximant of a Stieltjes
function, J. van Iseghem.
Interpolation between sequence transformations, E.J. Weniger.
Requests for sample copies and orders are to be sent to J.C. Baltzer AG,
fax: +41-61-692 42 62, e-mail: firstname.lastname@example.org
From: Anastassiou <ANASTASG@hermes.msci.memst.edu>
Date: 25 Mar 93 12:15:07 CDT
Subject: Contents, Journal of Computational and Applied Mathematics
Journal of Computational and Applied Mathematics
vol. 45, number 3, 3 May 1993
S.K. Bhatta and K.S. Sastri
Symmetric spline procedures for boundary value problems with mixed
C. Carstensen and M.S. Petkovic
On iteration methods withoug dervatives for the simulataneous
determination of polynomial zeros
H. Le Ferrand
Vector orthogonal polynomials and matrix series
M.N. Mikhail and M.R. El-Tantawy
Effect of the medium viscosity on soound propagation and attenuation in ducts
A BIE method for a nonlinear BVP
Error estimates in projexctive solutions of the Radon equation
J.A. Ferreira dn P. De Oliveira
Convergence properties of numerical discretizations and regridding methods
Some further results on the Hubbell rectangular source integral
Optimal difference schemes for 2-D transport problems
E.H. Kaufman Jr and G.D. Taylor
Coupled constrained rational approximation
Nguyen huu Cong
Note on the performance of direct and indirect Runge- Kutta-Nystrom methods
P.W. Hemker and J.-A. Desideri
Convergence behavious of defect correction for hyperbolic equations
M.K. Oberle, S.L. Scott, G.T. Gilbert, R.L. Hatcher and D.F. Addis
Mellin transforms of a generalization of Legendre polynomials
End of NA Digest