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Spline Toolbox™ 3
User’s Guide
Carl de Boor
How to Contact The MathWorks:
www.mathworks.com Web
comp.soft-sys.matlab Newsgroup
www.mathworks.com/contact_TS.html Technical support
suggest@mathworks.com
Product enhancement suggestions
bugs@mathworks.com Bug reports
doc@mathworks.com Documentation error reports
service@mathworks.com Order status, license renewals, passcodes
info@mathworks.com Sales, pricing, and general information
508-647-7000 (Phone)
508-647-7001 (Fax)
The MathWorks, Inc.
3 Apple Hill Drive
Natick, MA 01760-2098
For contact information about worldwide offices, see the MathWorks Web site.
Spline Toolbox
™ User’s Guide
© COPYRIGHT 1990–2008 by C. de Boor and The MathWorks, Inc.
The software described in this document is furnished under a license agreement. The software may be used
or copied only under the terms of the license agreement. No part of this manual may be photocopied or repro-
duced in any form without prior written consent from The MathWorks, Inc.
FEDERAL ACQUISITION: This provision applies to all acquisitions of the Program and Documentation by,
for, or through the federal government of the United States. By accepting delivery of the Program or
Documentation, the government hereby agrees that this software or documentation qualifies as commercial
computer software or commercial computer software documentation as such terms are used or defined in
FAR 12.212, DFARS Part 227.72, and DFARS 252.227-7014. Accordingly, the terms and conditions of this
Agreement and only those rights specified in this Agreement, shall pertain to and govern the use,
modification, reproduction, release, performance, display, and disclosure of the Program and Documentation
by the federal government (or other entity acquiring for or through the federal government) and shall
supersede any conflicting contractual terms or conditions. If this License fails to meet the government's
needs or is inconsistent in any respect with federal procurement law, the government agrees to return the
Program and Documentation, unused, to The MathWorks, Inc.
Trademarks
MATLAB and Simulink are registered trademarks of The MathWorks, Inc. See
www.mathworks.com/trademarks for a list of additional trademarks. Other product or brand
names may be trademarks or registered trademarks of their respective holders.
Patents
The MathWorks products are protected by one or more U.S. patents. Please see
www.mathworks.com/patents for more information.
Revision History
March 1990 First printing New for Version 1.0
November 1992 Second printing Revised for Version 1.1
January 1998 Third printing Revised for Version 2.0 (Release 10)
January 1999 Online only Revised for Version 2.0.1 (Release 11)
September 2000 Fourth printing Revised for Version 3.0 (Release 12)
May 2001 Fifth printing Minor revision for Version 3.0 (Release 12.1)
December 2001 Online only Revised for Version 3.1
February 2003 Sixth printing Revised for Version 3.2 (Release 13)
June 2004 Seventh printing Revised for Version 3.2.1 (Release 14)
June 2005 Eighth printing Minor revision for Version 3.2.1
September 2005 Online only Minor revision for Version 3.2.2 (Release 14SP3)
March 2006 Ninth printing Revised for Version 3.3 (Release 2006a)
September 2006 Online only Minor revision for Version 3.3.1 (Release 2006b)
September 2006 Tenth printing Version 3.3.1
March 2007 Online only Revised for Version 3.3.2 (Release 2007a)
May 2007 Eleventh printing Version 3.3.2
September 2007 Online only Revised for Version 3.3.3 (Release 2007b)
March 2008 Online only Revised for Version 3.3.4 (Release 2008a)
October 2008 Online only Revised for Version 3.3.5 (Release 2008b)
v
Contents
1
Getting Started
Product Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-2
MATLAB® Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-4
Expected Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-5
Technical Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-6
Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-6
Naming Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-6
Using Spline Toolbox™ Functions . . . . . . . . . . . . . . . . . . . . . . . 1-7
2
Some Simple Examples
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-2
Cubic Spline Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-3
Periodic Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-4
Other End Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-5
General Spline Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . 2-6
Knot Choices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-8
Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-9
Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-11
Using the Spline Fits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-12
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