Numerical Methods Qualification Exam Problems and Solutions (University of Maryland)/August 2009

Problem 1Edit

Let   be a real symmetric matrix of order   with   distinct eigenvalues, and let   be such that   and the inner product   for every eigenvector   of  .

Problem 1aEdit

Let   denote the space of polynomials of degree at most  . Show that


defines an inner product on  , where the expression on the right above is the Euclidean inner product in  

Solution 1aEdit



Linearity of 1st ArgumentEdit




Positive DefinitenessEdit



We also need to show that   if and only if  .

Forward Direction (alt)Edit

Suppose  . It suffices to show  . However, this a trivial consequence of the fact that   (which is clear from the fact that   for   with degree less than   and that   does not lie in the orthogonal compliment of any of the   distinct eigen vectors of  ).

Forward DirectionEdit

Claim: If  , then  .

From hypothesis


where   are the orthogonal eigenvectors of   and all   are non-zero


Notice that   is a linear combination of  , the coefficients of the polynomial  , and the scaling coefficient   of the eigenvector e.g.  

Since   and  , this implies  .

Reverse DirectionEdit

If  , then  

Problem 1bEdit

Consider the recurrence


where the   and   are scalars and  . Show that  , where   is a polynomial of degree  

Solution 1bEdit

By induction.

Base CaseEdit



Induction StepEdit






where   (respectively  ) has degree   (respectively  ). Then for  


which is as desired.

Problem 1cEdit

Suppose the scalars above are such that   and   is chosen so that  . Use this to show that that the polynomials in part (b) are othogonal with respect to the inner product from part (a.

Solution 1cEdit

Since   and  , it is equivalent to show that   for  .



it is then sufficient to show that


Claim  Edit

By induction.

Base CaseEdit


Induction StepEdit




Claim  Edit

By induction.

Base CaseEdit



Induction StepEdit





Problem 2Edit

Consider the n-panel trapezoid rule   for calculating the integral   of a continuous function  ,



Problem 2aEdit

Find a piecewise linear function   such that


for any continuous function   such that   is integrable over [0,1]. Hint: Find   by applying the fundamental theorem of calculus to  .

Solution 2aEdit

Rewrite given equation on specific intervalEdit

For a specific interval  , we have from hypothesis


Distributing and rearranging terms gives


Apply HintEdit

Starting with the hint and applying product rule, we get


Also, we know from the Fundamental Theorem of Calculus


Setting the above two equations equal to each other and solving for   yields


Choose G'(t)Edit

Let  . Therefore since   is linear


By comparing equations (1) and (2) we see that



Plugging in either   or   into equation (3), we get that




Problem 2bEdit

Apply the previous result to  ,  , to obtain a rate of convergence.

Solution 2bEdit

Problem 3Edit

Let   denote the set of all real-valued continuous functions defined on the closed interval   be positive everywhere in  . Let   be a system of polynomials with   for each  , orthogonal with respect to the inner product


For a fixed integer  , let   be the   distinct roots of   in  . Let


be polynomials of degree  . Show that


and that


Hint: Use orthogonality to simplify  

Solution 3aEdit


Solution 3bEdit





Since   is a polynomial of degree   for all  ,   is a polynomial of degree  .

Notice that   for   where   are the   distinct roots of  . Since   is a polynomial of degree   and takes on the value 1,   distinct times