Todd Will
UW-La Crosse

Bill and Ted's misadventure.

Let 

• a be the number of grams of A, 
• b be the number of grams of B, 
• c be the number of grams of C. 

"Shoot!", says Bill. 

"We've got four equations and only three unknowns. 

Everyone knows you only need three equations to solve for three unknowns. 

Let's just delete the last equation." 
 

Ted agreed with Bill and so they decided to solve the system: 

"Hold on", says Ted. "We can only solve this equation if the coefficient matrix is invertible." 

After a lot of work Ted computes 

. 
 

Seeing that the determinant is non-zero, Bill and Ted feel confident in computing 

. 

Bill and Ted were pleased with their results, but they realized that the numbers they recorded on the right hand side of their equations, ,  were only accurate to the nearest tenth. 

They decided to round their numbers and try the calculation again: 

"Noooooooooo!", screamed Bill, "how could this happen?" 

"We've changed our matrix equation only a little but now we get a wildly different answer." 

Explain to Bill and Ted what went wrong.

An SVD for A is . 

Since A has three non-zero stretch factors, the rank of A is three and A is invertible. 

But that smallest singular value is so small that A is nearly a rank two matrix. 

This spells trouble. 

Using the SVD method you can compute 

. 

Now let y be the original right hand side and let y' be the rounded right hand side. 

You need to explain why  is so big. 

Well, . 

Now the matrix  stretches vectors parallel to  by a factor of 5000. 

So if  is parallel to , then . 

This says that changes in the right hand side can get magnified up to 5000 times when you solve for x. 

Explain to Bill and Ted how to fix their problem.

In the real world you want all the data you can get. 

Here's the original system with the fourth equation returned. 
 

Now a reduced SVD for the coefficient matrix is 

This matrix is a fully robust rank 3 matrix. 

Since the singular values of A are each roughly 100, the singular values of the pseudo-inverse, , will each be roughly . 

This tells you that if  is the original right hand side 
and  is the rounded right hand side, 
then . 
 

So the theory tells you that rounding y to y' won't affect the solution to the system. 

That's the theory, now here are the facts: 

Value for a,b,c using the original right hand side. 

Value for a,b,c using the rounded right hand side. 

The theory holds up! 

Theorem: If A has singular values , 
then .

Then the fundamental equations tell you . 

So  

. 
 
 

(1) If  is an orthonormal basis, then 
 

---

Exercises 

 

and then solved for . 
 

Give an upper bound on . 
 

Hint:   

Compute the condition number for 
 

• the original coefficient matrix........ 
  
  
  
• the coefficient matrix with the last row deleted....... 

A matrix with a large condition number is called ill-conditioned and often displays extreme sensitivity to rounded like Ted and Bill saw.