Is There a Simpler Method to Solve Complex Recurrence Relations?

[Jamin2112]

Homework Statement

Homework Equations

n/a

The Attempt at a Solution

I understand that I could make a REALLY complicated recurrence relation, one involving like 5 nested for loops, but I would prefer a better way of finding d₁, ..., d_n.


Is there a better way?

 

[micromass]

Uuh, what is the complete question?? All I can see is the question "Compute coefficients so that the polynomial..."

Maybe attach your pdf to here??

 

[Jamin2112]

Homework Statement

Homework Equations

n/a

The Attempt at a Solution

I understand that I could make a REALLY complicated recurrence relation, one involving like 5 nested for loops, but I would prefer a better way of finding d₁, ..., d_n.


Is there a better way?

Whoops!


" ... so that the polynomial satisfies the n conditions

p(t₁) = b₁ , ... , p(t_n) = b_n "

 

[micromass]

Is the recurrence THAT complicated?? I'm really surprised how you can find 5 nested loops for this problem. Care to explain your reasoning?

What you could do is calculate (theoretically) p(t₁),...,p(t_n), and see what you get. It shouldn't be that hard...

 

[Jamin2112]

Is the recurrence THAT complicated?? I'm really surprised how you can find 5 nested loops for this problem. Care to explain your reasoning?

What you could do is calculate (theoretically) p(t₁),...,p(t_n), and see what you get. It shouldn't be that hard...

I still think it's complicated.

We have

b₁ = d₁
b₂ = d₁ + d₂(t₂ - t₁)
b₃ = d₁ + d₂(t₃ - t₁) + d₃(t₃ - t₁)(t₃ - t₂)
b₄ = d₁ + d₂(t₄ - t₁) + d₃(t₄ - t₁)(t₄ - t₂) + d₄(t₄ - t₁)(t₄ - t₂)(t₄ - t₃)
.
.
.

And it got really messy when I tried to find a general formula for di in terms of d_i-1

 

[micromass]

Ah, yes, I see the problem

I still think it's complicated.

We have

b₁ = d₁
b₂ = d₁ + d₂(t₂ - t₁)
b₃ = d₁ + d₂(t₃ - t₁) + d₃(t₃ - t₁)(t₃ - t₂)
b₄ = d₁ + d₂(t₄ - t₁) + d₃(t₄ - t₁)(t₄ - t₂) + d₄(t₄ - t₁)(t₄ - t₂)(t₄ - t₃)

.
Express this as a system of linear equations, and thus as matrices. MATLAB is really good in solving these!

And it got really messy when I tried to find a general formula for di in terms of d_i-1

Don't worry with a general formula. I suppose you'll only need to solve it for specific values...

 

[Jamin2112]

Ah, yes, I see the problem


.
Express this as a system of linear equations, and thus as matrices.

Ooooooooooohhhhhhh! HOW DID I NOT SEE THAT BEFORE?


[URL]http://www.gifbin.com/bin/1233445870_ae19b02.gif[/URL]

 

[Jamin2112]

Ah, yes, I see the problem


.
Express this as a system of linear equations, and thus as matrices. MATLAB is really good in solving these!



Don't worry with a general formula. I suppose you'll only need to solve it for specific values...

Is this the sort of set-up I'll have?

 

[micromass]

Yes, it would be something like that that you'll get.