Jonathan Dorfman

Current Important Information

Textbook and Matlab Information

• Required Textbook: Numerical Methods using Matlab by Mathews and Fink
• Recommended Textbook: Numerical Analysis by Burden and Faires
• Some online resources for Textbook
  • Mathews' web page
  • Numerical Modules (closely follows each Chapter of the text)
  • Numerical Modules Matlab Source (also follows each Chapter of the text)
• Some (of the endless) online resources for Matlab
  • Matlab Primer by Kermit Sigmon (hmtl, pdf)
  • Tutorial from Mathworks
  • Tutorial from MIT
  • Tutorial from Univ. Florida
  • Links
  • More Links

Miscellaneous Information

• Detailed Schedule for Spring 2005

Old News

• For Problem 11.1, you can use the B-spline recurrence relation and do the algebra by hand to specify the exact cubic defined over each of the intervals [0, 0], [0, 1], [1, 3], [3, 6].
  Another approach (which can be used to check your work) is to use a modified CreateBkN.m which takes as a parameter the number of subdivisions between each knotpoint (instead of a discretization parameter 'h' for the entire domain). By passing N=4 you can then pass the 4 points X(1:4),Bk4(1:4) to the canned Lagrange Interpolation function by Mathews and Fink Lagrange.m to retrieve the array of cubic coefficients over [t1,t2]. Similarly, passing X(4:7),Bk4(4:7) will get the cubic coefficients over [t2,t3]. Summarizing, the relevant functions are:
  • Lagrange.m (arrays passed as column vectors)
  • CreateBkN.m (knots passed as row vector)
  • BSpline.m
  • CreateOmega.m
  • FormatPoly.m (converts coefficient array to formatted polynomial - compatible with output from Lagrange.m.)
  Your code will need to account for the knot multiplicty (t1=t2) - this is best done by specifically checking for multiple knots, but replacing the knot sequence with [0, 0+eps, 1, 3, 6] will give you an idea of what results to expect.
• For Problem 11.2, the built-in Matlab function sort will be useful. It's most efficient to implement the check by using loops which kick out as soon as a failed condition is encountered, but if you prefer grossly inefficient one-line Matlab code, then you can use the following:
  • Remember that a conditional expression applied to arrays like x > y returns an array of 0's and 1's depending on whether the condition holds true for each entry.
  • Use the built-in function nnz to check if an array of 0-1's is all 1's.
• For Problem 11.3, it is preferable to precompute the binomial coefficients instead of computing them (over and over again) at runtime. By loading a datafile the binomial coefficients can be retrieved at runtime by a simple function that performs an efficient table lookup.
• Main code for Problem 11.3 should be structured as follows:

  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  %tell Binomial function to load the Binomial.mat datafile
  Binomial('init');
  
  %specify t-discretization for Bezier curve
  h	= .001;
  
  figure(31)
  CP	= [7 4 2 0; 0 -3 -1 0]';
  xy	= Bezier(CP, h);
  plot(CP(:,1),CP(:,2),'-',xy(:,1),xy(:,2),'.');
  ...etc...
  

• The source file Binomial.m and the data file Binomial.mat need to be included in your project.
• Using the above code you need only implement Bezier.m:

  function	xy = Bezier(CP,h)
  
  %on entry, CP=[column of x-coordinates of control polygon vertices,
  %   		column of y-coordinates of control polygon vertices]
  % h=t-discretization
  %Returns xy=discretizations of Bezier curve
  % xy(:,1)=x-coordinates, xy(:,2)=y-coordinates
  
  sizeCP	= size(CP);
  N	= sizeCP(1);	%degree of Bezier polynomial=(N-1)
  BC	= Binomial(N);	%returns array of (N-1)!/(j!(N-1-j)!) length=N
  ...
  return
  

• In the second part of 10.1, use "x=1.95" in addition to the values specified for "t, a, & k".
• When debugging 10.2 remember the vertices of the refined control polygon must lie on the original control polygon.
• Wrapper for your code in problem 10.3 which should show the control polygons converging to F
  (and so can also be used for debugging)
  
  • CodeFragment (NOTE: CreateBk(t,k,h) expects the knot-sequence 't' to be passed as a row vector)
  • CreateBk.m
  • BSpline.m
  • CreateOmega.m
• The following code dirich.m (extracted from Mathews' website, and modified to correct a bug in the initial values at the corners and also to suppress the flipud(U) at the end) may be useful for Assignment #9. Also use "[X, Y] = meshgrid(x,y), surf(X,Y,U)" so (X,Y)-values are used instead of array indices.
• For additional information on solving Laplace's equation (coded in Mathematica) see Mathews' module on Elliptic PDEs.
• For additional information on splines (coded in Mathematica) see Mathews' module on B-splines.
  Note that he remarks in these notes: "CAVEAT: If you have unequally spaced points, then this is construction does not apply and construction of the cubic spline require a more cumbersome algorithm because each piecewise cubic will need to be individually crafted in order to meet all the conditions for a cubic spline." - deBoor's treatment does not have this limitation.
• Dropping and Adding information (after Telebears ends):
  • March 11, 2005 is the deadline to add or drop without the dean's approval.
  • form
  • instructions.
• I am cancelling the Thursday Office Hour. Instead, I encourage students to email me to arrange an appointment if you need to see me outside of class.
• I created a utility function which you can use to help format your performance output from Homework Assignment #3:
• I created a couple of utility functions which you can use to help format your output from Homework Assignment #2:
• I prepared a document giving some background material related to norms.
• Note the new hours for Math 98:
  Monday 1:30-3 pm and Wednesday 4:30-6 pm
  
• There is a short course in Matlab offered by the Math Department called Math98. It meets 3 hours/week in the Computer Lab: TuTh 2-3:30 pm and Mon 1:30-3 pm/Wed 4:30-6 pm. Be aware of the following:
  • In order to attend Math98 you must be Enrolled (1 Unit, P/NP)
• I've scheduled the second discussion to take place on Tuesday 1-2 in the Computer Lab.
• I created a Figures page for those unable to handle postscript files. (Note: This is just meant for viewing - if you save the picture directly from your browser it could end up being a 3MB .bmp-file)
• Scheduler tool for submitting available time slots.
• ScheduleAnding tool for determining mutually available time slots.
• Other UCB Courses using Matlab
  • Engineering 77
    • Textbook: Matlab Programming by Kuncicky
    • Textbook: Mastering Matlab 7 by Hanselman & Littlefield
  • Engineering 170B and 180A
    • Textbook: Numerical Methods using Matlab by Lindfeld & Penny