Determining polynomial coefficients by known derivatives

Omur (view profile)

on 18 Oct 2012
Latest activity Commented on by Star Strider

Star Strider (view profile)

on 8 Jul 2017
Accepted Answer by Dr. Seis

Dr. Seis (view profile)

Hello community, I'm trying to create a cubic polynomial function, I know these: y(0)=1 y(30)=0.5 dy(0)/dx=-1 dy(30)/dx=-1 Can I determine the values for coefficients ?

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Answer by Dr. Seis

Dr. Seis (view profile)

on 18 Oct 2012
Edited by Dr. Seis

Dr. Seis (view profile)

on 18 Oct 2012

Yes. Just in case this is homework, I will give an example to get you started.
Let's say you have the case below (I will start with the case where we do not have derivatives at a location).
We are trying to solve for A, B, and C
y(x) = A*x^2 + B*x + C
Knowns:
x1 = 0; y1 = 1;
x2 = 3; y2 = 6;
x3 = 7; y3 = 7;
So we can construct an inverse problem of the form G * m = d, where:
G = [x1^2,x1,1; x2^2,x2,1; x3^2,x3,1];
d = [y1; y2; y3];
We determine m (equal to [A; B; C] in our example) by:
m = G\d;
And we get:
plot(-1:10,polyval(m,-1:10),'r-', [x1,x2,x3],[y1,y2,y3],'k+') Dr. Seis

Dr. Seis (view profile)

on 18 Oct 2012
Almost... since you will need to expand to a 3-order polynomial, I will give you the G for a 2-order (which would give you the plot immediately above if dydx2 = +1). In your case, your y and y' info are at the same 2 x locations... so:
| x1^2, x1, 1 | | A | | y1 |
| x2^2, x2, 1 | | B | = | y2 |
| 2*x1, 1, 0 | | C | | dydx1 |
| 2*x2, 1, 0 | | dydx2 |
You still need to have contribution from your "B", which is why the second column of the last two rows have the 1. Once you expand to a 3-order polynomial for your data, you should get something like: Dr. Seis

on 18 Oct 2012
Omur

Omur (view profile)

on 18 Oct 2012
Thank you so much !

Answer by Suzie Oman

Suzie Oman (view profile)

on 8 Jul 2017

This answer doesn't incorporate the derivative. How would this be solved taking into account the derivative?
What about in 3 dimensions?

Star Strider

Star Strider (view profile)

on 8 Jul 2017
See this Comment (link).