Direct Method of Interpolation: Cubic Interpolation - Part 1

[agravesf]

I suspect that is it, everytime the power of the polinomial increases we then define one more point as part of the function. For the quadratic you used two points, but I think that i will need to use 3 for the cubic and so on.

This will then give me 15 equations in step 1), which will then allow me to assume in step 3) that my process starts with a qquadratic function.

[agravesf]

2) now we have left 10 euqations to find

if we now make the derivatives equal at the interior points we will be able to get 4 equations

3) if we now force two of the functions parameters say a1=0 and b1=0, such that the first function is linear

Now I still have left 4 equations to be found... !!!!!

So where am I going wrong?

This has made me question step 1) and if it is in fact every two points that we have a defined function...

...continue next post...

[agravesf]

YET WHEN in the spline method we say that:

1) every two point we have one function, hence #functions = n

# functions * (polynomial power+1) = unknowns

On the cubic case we therefore have: n*4, in your exmaple n=5, therefore 20 unknowns.

If every two points we have one function and each point can comply with two different functions the we are able to generate 2 equations per interior points and one equation per extreme points, we are able to get 10 equations.

...continue next post..

[agravesf]

I am struggling to derive this same solution (set of equations) for a cubic spline.

Ok, here is my doubt, it is more conceptually actually.

When we solve for local polynomials we need:

Linear: we need two points (x1,y1) and (x2,y2) thus 2 equations.

Quadratic: we need at least three data points (x1,y1), (x2,y2), (x3,y3), thus 3 equations.

Cubic: we need at least four data points (x1,y1), (x2,y2), (x3,y3), (x4,y4), thus 4 equations

... continue in next post...

[numericalmethodsguy]

@agravesf The second order derivatives are also continuous at the interior data points.

6 points - 5 splines - 20 unknowns. Each spline goes thru two consecutive pts - 10 eqns, splines have 1st derivative continuous at interior pts - 4 eqns, splines have 2nd derivative continuous at interior pts - 4 eqns, first spline is quadratic and last spline is quadratic - 2 eqns. Total 20eqns.

[alfredo555]

Nice and easy example.

[LAnonHubbard]

This is really great. Not only am I getting information about what I really want to learn about (polynomial interpolation, splines, etc), I'm getting a chance to put into practice my previous learning (from Khan Academy and a 3D game programming book) about using matrices to solve systems of linear equations. What I learnt there has faded from memory a bit so I can hopefully pull it all together.

[Antikceo]

great videos

[staM034]

that's my guy !!!

[numericalmethodsguy]

@YourWorstNightmareDK May I suggest going to the numericalmethods(dot)eng(dot)u­sf(dot)edu website, click on keyword, click on Matrix Algebra. That will bring your weak side to a strong side.

[gefallt]

Wow, you did in 9 minutes what my teacher did in 2 weeks..........

[CrazyTacoooO]

Thank You very much

You've helped me a lot with this vid

I wish You'd be my teacher at the University =)

[Danny77uk]

Thankyou!