Re: Derivative of BernsteinBasis functions
- To: mathgroup at smc.vnet.net
- Subject: [mg118355] Re: Derivative of BernsteinBasis functions
- From: Bob Hanlon <hanlonr at cox.net>
- Date: Sun, 24 Apr 2011 08:24:47 -0400 (EDT)
{$Version, $ReleaseNumber}
{"8.0 for Mac OS X x86 (64-bit) (February 23, 2011)", 1}
g[x_] = -BernsteinBasis[5, 0, x] + 6 BernsteinBasis[5, 1, x] -
15 BernsteinBasis[5, 2, x] + 20 BernsteinBasis[5, 3, x] -
15 BernsteinBasis[5, 4, x] + 6 BernsteinBasis[5, 5, x];
Simplify this prior to taking derivative
g[x_] = FullSimplify[g[x]]
Piecewise[{{-1 + 7*x*(5 + 2*x*(-20 + 3*x*(20 + x*(-25 + 11*x)))),
0 <= x <= 1}}, 0]
g''[x]
Piecewise[{{Indeterminate,
x == 0 || x == 1}, {35*(-16 + 144*x - 360*x^2 + 264*x^3), 0 < x < 1}}, 0]
To show that this is equivalent to the result from version 7
h[x_] =
-20 BernsteinBasis[3, 0, x] + 30 (-4 BernsteinBasis[3, 0, x] -
4 (BernsteinBasis[3, 0, x] - BernsteinBasis[3, 1, x])) -
75 (4 (BernsteinBasis[3, 0, x] - BernsteinBasis[3, 1, x]) -
4 (BernsteinBasis[3, 1, x] - BernsteinBasis[3, 2, x])) +
100 (4 (BernsteinBasis[3, 1, x] - BernsteinBasis[3, 2, x]) -
4 (BernsteinBasis[3, 2, x] - BernsteinBasis[3, 3, x])) -
75 (4 (BernsteinBasis[3, 2, x] - BernsteinBasis[3, 3, x]) -
4 BernsteinBasis[3, 3, x]) + 120 BernsteinBasis[3, 3, x];
Reduce[g''[x] == h[x], x]
0 < x < 1 || x < 0 || x > 1
So they are equal for real x excluding the two indeterminate points.
Bob Hanlon
---- "Rob Y. H. Chai" <yhchai at ucdavis.edu> wrote:
=============
Hello everyone,
I'm trying to compute the second derivative of a function g[x_] that
involves a combination of Bernstein Basis functions. Mathematica version 7 was able to compute the derivative, but version 8 gave a problem.
In[1]:= {$Version,$ReleaseNumber}
Out[1]= {7.0 for Microsoft Windows (32-bit) (February 18, 2009),1}
In[2]:= g[x_] = -BernsteinBasis[5,0,x]+6 BernsteinBasis[5,1,x]-15
BernsteinBasis[5,2,x]+20 BernsteinBasis[5,3,x]-15 BernsteinBasis[5,4,x]+6
BernsteinBasis[5,5,x]
Out[2]= -BernsteinBasis[5,0,x]+6 BernsteinBasis[5,1,x]-15
BernsteinBasis[5,2,x]+20 BernsteinBasis[5,3,x]-15 BernsteinBasis[5,4,x]+6
BernsteinBasis[5,5,x]
In[3]:= D[g[x],{x,2}]
Out[3]= -20 BernsteinBasis[3,0,x]+30 (-4 BernsteinBasis[3,0,x]-4
(BernsteinBasis[3,0,x]-BernsteinBasis[3,1,x]))-75 (4
(BernsteinBasis[3,0,x]-BernsteinBasis[3,1,x])-4
(BernsteinBasis[3,1,x]-BernsteinBasis[3,2,x]))+100 (4
(BernsteinBasis[3,1,x]-BernsteinBasis[3,2,x])-4
(BernsteinBasis[3,2,x]-BernsteinBasis[3,3,x]))-75 (4
(BernsteinBasis[3,2,x]-BernsteinBasis[3,3,x])-4 BernsteinBasis[3,3,x])+120
BernsteinBasis[3,3,x]
But
{$Version,$ReleaseNumber}
{8.0 for Microsoft Windows (32-bit) (February 24, 2011),1}
D[g[x],{x,2}]
Gives an error. Any thoughts?
Thanks.
Rob Chai