add hermite polynomials computation and normalized

matlab/compute/HermitePhys.m

+
+% HermitePoly.m by David Terr, Raytheon, 5-10-04
+
+% Given nonnegative integer n, compute the 
+% Hermite polynomial H_n. Return the result as a vector whose mth
+% element is the coefficient of x^(n+1-m).
+% polyval(HermitePoly(n),x) evaluates H_n(x).
+
+
+function hk = HermitePoly(n)
+% Evaluate the normalized Hermite polynomial.
+if n==0 
+    hk = 1;
+elseif n==1
+    hk = [2 0];
+else
+    
+    hkm2 = zeros(1,n+1);
+    hkm2(n+1) = 1;
+    hkm1 = zeros(1,n+1);
+    hkm1(n) = 2;
+
+    for k=2:n
+        
+        hk = zeros(1,n+1);
+
+        for e=n-k+1:2:n
+            hk(e) = 2*(hkm1(e+1) - (k-1)*hkm2(e));
+        end
+        
+        hk(n+1) = -2*(k-1)*hkm2(n+1);
+        
+        if k<n
+            hkm2 = hkm1;
+            hkm1 = hk;
+        end
+        
+    end
+    
+end
\ No newline at end of file

matlab/compute/HermitePhys_norm.m

+% Normalized version of
+% HermitePoly.m by David Terr, Raytheon, 5-10-04
+
+% Given nonnegative integer n, compute the 
+% Hermite polynomial H_n. Return the result as a vector whose mth
+% element is the coefficient of x^(n+1-m).
+% polyval(HermitePoly(n),x) evaluates H_n(x).
+
+
+function hk = Hp(n)
+% Evaluate the normalized Hermite polynomial.
+if n==0 
+    hk = 1;
+elseif n==1
+    hk = [2 0];
+else
+    
+    hkm2 = zeros(1,n+1);
+    hkm2(n+1) = 1;
+    hkm1 = zeros(1,n+1);
+    hkm1(n) = 2;
+
+    for k=2:n
+        
+        hk = zeros(1,n+1);
+
+        for e=n-k+1:2:n
+            hk(e) = 2*(hkm1(e+1) - (k-1)*hkm2(e));
+        end
+        
+        hk(n+1) = -2*(k-1)*hkm2(n+1);
+        
+        if k<n
+            hkm2 = hkm1;
+            hkm1 = hk;
+        end
+        
+    end
+end
+% Normalization
+% hk = 1/sqrt(2^n*factorial(n))/sqrt(sqrt(pi))*hk;
+hk = 1/sqrt(2^n*factorial(n))/sqrt(sqrt(pi))*hk;