Bibliography

Peter Alfeld, A Survey of Zadunaisky's Device
Applied to Ordinary Differential Equations, M.Sc.
Thesis, The University Dundee, 1975, pp 1132.

Peter Alfeld, Correction in the Dominant Space: A
New Technique for the Numerical Solution of Certain
Stiff Initial Value Problems, Ph.D. Thesis, The
University Dundee, 1977, pp 1344.

Peter Alfeld and J.D. Lambert, Correction in the
Dominant Space: A numerical technique for a certain
class of stiff initial value problems, Math. Comp.
31 (1977), pp 922938.

Peter Alfeld, Inverse Linear Multistep Methods for
the Numerical Solution of Initial Value Problems of
Ordinary Differential Equations, Math. Comp. 33
(1979), pp 111124.

Peter Alfeld, An Improved Version of the Reduction
to Scalar CDS Method for the Numerical Solution of
Separably Stiff Initial Value Problems, Math. Comp.
33 (1979), pp 535539.

Peter Alfeld, A Special Class of Explicit Linear
Multistep Methods for the Correction in the Dominant
Space technique, Math. Comp. 33 (1979), pp
11951212.

Peter Alfeld, Two Devices for Improving the
Efficiency of Stiff ODE Solvers, in Proceedings of
the 1979 SIGNUM Meeting on Numerical Ordinary
Differential Equations, U. Ill. Dept. of Comp. Science
Rep. 791710, pp 241 to 243.

Peter Alfeld, A Method of Skipping the Transient
Phase in the Solution of Separably Stiff Ordinary
Initial Value Problems, Math. Comp. 35 (1980), pp
11731176.

Peter Alfeld, Fixed Point Iteration With Inexact
Function Values, Math. Comp. 38 (1982), pp 8798.

Peter Alfeld, A Reduce Algorithm
for the symbolic computation of Padé
approximants, manuscript.

F.C. Hoppensteadt, Peter Alfeld, and R.C. Aiken,
Numerical Treatment of Chemical Kinetics by Perturbation
and Projection Methods, in Modelling of Chemical
Reaction Systems, K.Ebert, P.Deuflhard, W. Jäger,
eds., SpringerVerlag, 1981, pp 3138.

K. Furlong, D. Chapman, and Peter Alfeld, Thermal
Constraints on the Geometry of Subduction: Tectonic
Implications, Journal of Geophysical Research, v
87, 1982, pp 17861802.

F.C. Hoppensteadt and Peter Alfeld, Explosion Mode
Analysis of an H_2O_2 Reaction, in R.C. Aiken
(ed.) Proceedings of The International Conference on
Stiff Systems, Park City, April 1214, 1982.

Peter Alfeld, Least Squares and Number Theory,
manuscript. I periodically assign a
term project
based on this idea.

Peter Alfeld and R.E. Barnhill, A Transfinite C^2
Interpolant over Triangles, Rocky Mountain Journal
of Mathematics, v 14 (1984), pp 1739.

Peter Alfeld, Two Discrete C^2 Interpolants,
Appendix of above reference.

Peter Alfeld and Bill Harris,
MICROSCOPE:,
A Software System for Multivariate Analysis.
MRC Technical Summary Report #2701, Mathematics Research
Center, University of WisconsinMadison, 1984, plus a
portable FORTRAN software package of
approximately 6000 lines of code, available from
netlib
or
my web page.

Peter Alfeld, A discrete C^1 interpolant for
tetrahedral data, Rocky Mountain Journal of
Mathematics, v 14 (1984), pp 516.

Peter Alfeld, Multivariate Perpendicular
Interpolation, SIAM Journal on Numerical Analysis,
v 22 (1985), pp 95106.

Peter Alfeld, Derivative Generation from
Multivariate Scattered Data by Functional Minimization
, Computer Aided Geometric Design J., v 2 (1985),
pp 281296. You may view a
dvi file
or
postscript file.

Peter Alfeld, A Trivariate CloughTocher
Interpolation Scheme, Computer Aided Geometric
Design J., v 1 (1984), pp 169181.

Peter Alfeld, A Bivariate C^2 CloughTocher Scheme
, Computer Aided Geometric Design J., v 1 (1984),
pp 257267.

Peter Alfeld, Triangular Extrapolation, MRC
Technical Summary Report #2707, Mathematics Research
Center, University of WisconsinMadison, 1984.

P. Alfeld, On the Dimension of Piecewise Polynomial
Functions, in D.E. Griffiths and G.A.Watson (ed.)
Numerical Aalysis, Pitman Research Notes in Mathematics
Series, No. 140, pp. 123, Proceedings of the Biennial
Dundee Conference on Numerical Analysis, June 2528,
1985, Langman Scientific and Technical. You may view a
dvi file
or
postscript file.

Peter Alfeld, A Case Study of Multivariate
Piecewise Polynomials, in ``Geometric Modeling'',
G. Farin (ed.), SIAM publication, 1987, pp. 149160.
(This paper is revised periodically to provide a record
of a growing set of examples. This is the
the newest version.

Peter Alfeld, Trivariate Adaptive Cubature,
Proceedings of the Fifth International Symposium on
Approximatioon Theory, College Station, Texas, January
1217, 1986, C. Chui, L.L. Schumaker and J.D. Ward
(ed.), Academic Press, 1986, pp. 231234.

Peter Alfeld, Bruce Piper, and L.L. Schumaker,
Minimally Supported Bases for Spaces of Bivariate
Piecewise Polynomials of Smoothness r and Degree d>
=4r+1, Computer Aided Geometric Design J 4 (1987),
pp. 105124.

Peter Alfeld and L.L. Schumaker, The Dimension of
Bivariate Spline Spaces of Smoothness r for Degree d>
=4r+1, J. Construct. Approx. Theory, Springer
Verlag, 1987, pp. 189197.

Peter Alfeld, Bruce Piper, and L.L. Schumaker, An
Explicit Basis for C^1 Quartic Bivariate Splines,
SIAM J. Num.Anal. 24 (1987), pp. 891911.

Peter Alfeld, Bruce Piper, and L.L. Schumaker,
Spaces of Bivariate Splines on Triangulations with Holes
, J. Approx. its Appl., v. 3 (1987), pp. 110.

Peter Alfeld, The Multivariate Spline Newsletter
, Published privately: Issue 1 (9/2/87), Issue 2
(2/21/88), Issue 3 (10/18/88).

Alfeld, P., Scattered Data Interpolation in Three
or More Variables, in Tom Lyche and Larry L.
Schumaker (eds), ``Mathematical Methods in Computer
Aided Geometric Design'', Academic Press, 1989, 134.

Alfeld, P., and Eyre, D.J, Algorithm 701,
Goliath , A Software System for the Exact
Analysis of Rectangular RankDeficient Sparse Rational
Linear Systems, ACM TOMS, 17 No. 4, December 1991,
519532.

Alfeld, P., L.L. Schumaker, and M. Sirvent, On
Dimension and Existence of Local Bases for Multivariate
Spline Spaces, Journal of Approximation Theory, 70
(1992), pp. 243264.

Peter Alfeld, David J. Eyre, and Larry L. Schumaker,
MachineAided Investigation of Multivariate Spline
Spaces, in C.K. Chui, L.L. Schumaker, and J.D.
Ward (eds), Approximation VI, Academic Press, 1989, 14.

Alfeld, P., and Sirvent, M., A Recursion Formula
for the Dimension of Superspline Spaces of Smoothness r
and Degree d > r2^k, W. Schempp and K. Zeller
(eds), Approximation Theory V, Proceedings of the
Oberwolfach Meeting, February 1218, 1989, Birkhä
user Verlag, pp. 18.

Alfeld, P., and Schumaker, L.L., 1989, On the
Dimension of Bivariate Spline Spaces of Smoothness r and
Degree d=3r+1, Numer. Math. 57, 651661 (1990).

Alfeld, P., and David Eyre, The Exact Analysis of
Sparse Rectangular Linear Systems, ACM TOMS, 17 No.
4, December 1991, 502518.

Alfeld, P., and Sirvent, M.,, The Structure of
Multivariate Superspline Spaces of High degree,
Math. Comp. 57 (1991), pp 299308.

Alfeld, P., L. L. Schumaker, and W. Whiteley, The
generic dimension of the space of C^1 splines of degree
d>= 8 on tetrahedral decompositions, SIAM JNA,
v. 30, pp. 889920, 1993. You may view a
dvi file
or
postscript file.

Alfeld, P., Upper and Lower Bounds on the Dimension
of Multivariate Spline Spaces, SIAM JNA, v.~33, No.
2, pp. 571588, April 1996. You may view a
dvi file
or
postscript file.

Alfeld, P., M. Neamtu, and L.L. Schumaker,
BernsteinBézier Polynomials on Spheres and
SphereLike Surfaces., CAGD Journal 13 (1996),
333349. You may view a
dvi file
or
postscript file.
Click here
to view otherwise unpublished graphs
of the BernsteinBézier Polynomials on the
Sphere.

Johnson, C., and Alfeld, P., Computational
Engineering and Science at the University of Utah,
IEEE Computational Science and Engineering, Fall 1994,
pp. 710.

Alfeld, P., M. Neamtu, and L.L. Schumaker,
Dimension and Local Bases of Homogeneous Spline Spaces
, SIAM J. Mathematical Analysis, v. 27, No. 5,
pp. 14821501, September 1996. You may view a
dvi file
or
postscript file.

Alfeld, P., M. Neamtu, and L.L. Schumaker,
Circular BernsteinBézier Polynomials, in
Mathematical Methods in CAGD, M. Daehlen, T. Lyche, and
L. L. Schumaker (eds), Vanderbilt University Press,
1995, 110. You may view a
dvi file
or
postscript file.

Alfeld, P., M. Neamtu, and L.L. Schumaker, Fitting
Scattered Data on SphereLike Surfaces using Spherical
Splines, Journal of Computational and Applied
Mathematics, 73 (1996), 543. You may view a
dvi file
or
postscript file.
Click here
to view otherwise unpublished examples
of our interpolants generated by some 16,000 lines of
code whose development took the three of us about 2
years.
 Alfeld, P., and L.L. Schumaker
Nonexistence of Starsupported
Spline Bases SIAM J. Math. Anal. 31 (2000), 455465.
 Alfeld, P, Bivariate Splines and Minimal Determining Sets,
Journal of Computational and Applied Mathematics, 119 (2000), 1327.
 Alfeld, P., and L.L. Schumaker,
Smooth MacroElements Based on CloughTocher Triangle Splits,
Numer. Math. 90 (2002), 597616.
 Alfeld, P., and L.L. Schumaker,
Smooth MacroElements Based on
PowellSabin Triangle Splits, Adv. Comp. Math., 16 (2002), 2946.
 Alfeld, P., and L.L.~Schumaker, Upper and Lower Bounds on the
Dimension of Superspline Spaces, Constructive Approximation
19 (2003), 145161.
 Alfeld, P., and L.L.~Schumaker, A C^{2} Trivariate
MacroElement Based on the CloughTocher Split of a Tetrahedron,
CAGD journal, 22 (2005), pp. 710721.
 Alfeld, P., and L.L.~Schumaker,A C^{2} Trivariate MacroElement
Based on the WorseyFarin Split of a Tetrahedron, SIAM Journal on
Numerical Analysis, 43 (2005), No. 4, pp. 17501756.
 Alfeld, P., and L.L.~Schumaker, A Trivariate DoubleCloughTocher
MacroElement, in Approximation Theory XI: Gatlinburg 2004, C. Chui,
M. Neamtu, and L. L. Schumaker (eds), Nashboro Press (Brentwood),
2005, 114.
 Alfeld, P., and L.L.~Schumaker, Bounds on the Dimensions of
Trivariate Spline Spaces,
submitted for publication, Advances in Computational Mathematics, Springer Verlag, DOI
10.1007/s1044400790516, 2007.
 Alfeld, P., and T. Sorokina; Two Tetrahedral C1 Cubic Macro Elements; Journal of
Approximation Theory, 157 (2009), 53.69, DOI: 10.1016/j.jat.2008.07.001.
 Alfeld, P., L.L. Schumaker, and T. Sorokina; Two Condensed MacroElements with
Full Approximation Power ; Advances in Comp. Math., 32 (2010), pp. 381391.
 [58] Alfeld, P.; Many Formulas; Journal of the Oughtred Society, v. 18, No. 2, 2009,
pp. 1821.
Keywords for this page: multivariate splines, spline
spaces, dimensions, interpolation, approximation,
interpolation on the sphere, homogeneous splines,
triangulations, finite elements, spherical splines, circular
splines, spherelike surfaces, tetrahedra, tetrahedral
decompositions, spline, dimension, tetrahedron, Bernstein
Polynomials, BernsteinBezier form, ODEs, stiff ODEs,
separably stiff ODEs, explosion mode analysis, nested
iteration, fixed point iteration.
Last revised: [19Feb2012]
Go to Peter Alfeld's Home Page.