def hermite_gauss_integral(m, n, x0, x1, stepsize=0.01):
"compute integral(h[m](x}*h[n](x)*exp(-x^2),{x,x0,x1})"
if m>n:
m,n = n,m #always make m<=n
xmax=max(abs(x0), abs(x1))
if not integral_cache.has_key((m,n)) or integral_cache[(m,n)][0] < xmax:
#just use Simpson's rule for this
#first, load up cache
hermite_gauss(m,xmax)
hermite_gauss(n,xmax)
xm, xa1, ya1, y2a1 = hermite_cache[m]
xm, xa2, ya2, y2a2 = hermite_cache[n]
stepcount=int(2.0*xmax/stepsize)
if stepcount%2==0:
stepcount=stepcount+1 #always odd for Simpson
stepsize=2*xmax/(stepcount-1) #exact step size
xlist=Numeric.array(range(stepcount),Numeric.Float)*stepsize - xmax
y=spline.splint(xa1, ya1, y2a1, xlist)*spline.splint(xa2, ya2, y2a2, xlist)
hmn2=spline.spline(xlist, y)
yint=Numeric.cumsum(y[0:-2:2]+y[2::2]+4.0*y[1::2])*(stepsize/3.0)
yint=Numeric.concatenate( ( (0,), yint )) #first element of integral is 0
yi2=spline.spline(xlist[::2], yint)
integral_cache[(m,n)]=(xmax, xlist[::2], yint, yi2, stepsize, xlist, y, hmn2)
xmax, xa, ya, y2a, stepjunk, xjunk, yjunk, hmn2junk=integral_cache[(m,n)]
iv1, iv2=spline.splint(xa, ya, y2a, (x0, x1) )
return iv2-iv1
def hermite_gauss_integral(m, n, x0, x1, stepsize=0.01):
"compute integral(h[m](x}*h[n](x)*exp(-x^2),{x,x0,x1})"
if m>n:
m,n = n,m #always make m<=n
xmax=max(abs(x0), abs(x1))
if not integral_cache.has_key((m,n)) or integral_cache[(m,n)][0] < xmax:
#just use Simpson's rule for this
#first, load up cache
hermite_gauss(m,xmax)
hermite_gauss(n,xmax)
xm, xa1, ya1, y2a1 = hermite_cache[m]
xm, xa2, ya2, y2a2 = hermite_cache[n]
stepcount=int(2.0*xmax/stepsize)
if stepcount%2==0:
stepcount=stepcount+1 #always odd for Simpson
stepsize=2*xmax/(stepcount-1) #exact step size
xlist=Numeric.array(range(stepcount),Numeric.Float)*stepsize - xmax
y=spline.splint(xa1, ya1, y2a1, xlist)*spline.splint(xa2, ya2, y2a2, xlist)
hmn2=spline.spline(xlist, y)
yint=Numeric.cumsum(y[0:-2:2]+y[2::2]+4.0*y[1::2])*(stepsize/3.0)
yint=Numeric.concatenate( ( (0,), yint )) #first element of integral is 0
yi2=spline.spline(xlist[::2], yint)
integral_cache[(m,n)]=(xmax, xlist[::2], yint, yi2, stepsize, xlist, y, hmn2)
xmax, xa, ya, y2a, stepjunk, xjunk, yjunk, hmn2junk=integral_cache[(m,n)]
iv1, iv2=spline.splint(xa, ya, y2a, (x0, x1) )
return iv2-iv1
def interpolate(self, actuators, width, int_act):
self.calls = self.calls + 1
n_act = numpy.shape(actuators)[0]
xinterp = numpy.zeros((width, n_act))
yinterp = numpy.zeros((width, width))
x = numpy.arange(n_act)
xint = numpy.arange(0, width) / float(int_act)
for row in range(0, n_act):
y = actuators[:, row]
xinterp[:, row] = spline.splint(x, y, xint)
# splint is Python coding of a Numerical Recipies function
x = numpy.arange(n_act)
xint = numpy.arange(0, width) / float(int_act)
for col in range(0, width):
y = xinterp[col, :]
yinterp[col, :] = spline.splint(x, y, xint)
return yinterp
def hermite_gauss_matrix_element(m, n, x0, x1, f, stepsize=0.01):
"compute integral(h[m](x}*h[n](x)*exp(-x^2)*f(x),{x,x0,x1}) : f should be able to be evaluated with a vector x"
if m>n:
m,n = n,m #always make m<=n
xmax=max(abs(x0), abs(x1))
if not integral_cache.has_key((m,n)) or integral_cache[(m,n)][0] < xmax:
hermite_gauss_integral(m,n,x0,x1) #preload cache with raw data
jxmax, jxa, jya, jy2a, jstepsize, xa, hmn, hmn2=integral_cache[(m,n)]
stepcount=int((x1-x0)/stepsize)
if stepcount%2==0:
stepcount=stepcount+1 #always odd for Simpson
stepsize=(x1-x0)/(stepcount-1) #exact step size
xlist=Numeric.array(range(stepcount),Numeric.Float)*stepsize + x0
y=spline.splint(xa, hmn, hmn2, xlist)*f(xlist)
yint=Numeric.sum(y[0:-2:2]+y[2::2]+4.0*y[1::2])*(stepsize/3.0)
return yint
def hermite_gauss_matrix_element(m, n, x0, x1, f, stepsize=0.01):
"compute integral(h[m](x}*h[n](x)*exp(-x^2)*f(x),{x,x0,x1}) : f should be able to be evaluated with a vector x"
if m>n:
m,n = n,m #always make m<=n
xmax=max(abs(x0), abs(x1))
if not integral_cache.has_key((m,n)) or integral_cache[(m,n)][0] < xmax:
hermite_gauss_integral(m,n,x0,x1) #preload cache with raw data
jxmax, jxa, jya, jy2a, jstepsize, xa, hmn, hmn2=integral_cache[(m,n)]
stepcount=int((x1-x0)/stepsize)
if stepcount%2==0:
stepcount=stepcount+1 #always odd for Simpson
stepsize=(x1-x0)/(stepcount-1) #exact step size
xlist=Numeric.array(range(stepcount),Numeric.Float)*stepsize + x0
y=spline.splint(xa, hmn, hmn2, xlist)*f(xlist)
yint=Numeric.sum(y[0:-2:2]+y[2::2]+4.0*y[1::2])*(stepsize/3.0)
return yint
def hermite_gauss(order,x, cache_key=None): "compute h[order](x)*exp(-x^2/2). x may be an array, and cache_key should be a unique identifier for x if cacheing is desired." if type(x) is not type(1.0): xmax=max(abs(x)) else: xmax=abs(x) if (not hermite_cache.has_key(order)) or hermite_cache[order][0] < xmax: q=generate_table(order, max(1.5*xmax, hermite_x_bound)) hermite_cache[order]=q if cache_key and hermite_cache.has_key((order, cache_key)): hermite_cache[(order, cache_key)][0]=time.time() #update time stamp return hermite_cache[(order, cache_key)][1] else: xmax, xa, ya, y2a=hermite_cache[order] y= spline.splint(xa, ya, y2a, x) if cache_key: hermite_cache[(order,cache_key)]=[time.time(), y] return y
def hermite_gauss(order,x, cache_key=None): "compute h[order](x)*exp(-x^2/2). x may be an array, and cache_key should be a unique identifier for x if cacheing is desired." if type(x) is not type(1.0): xmax=max(abs(x)) else: xmax=abs(x) if (not hermite_cache.has_key(order)) or hermite_cache[order][0] < xmax: q=generate_table(order, max(1.5*xmax, hermite_x_bound)) hermite_cache[order]=q if cache_key and hermite_cache.has_key((order, cache_key)): hermite_cache[(order, cache_key)][0]=time.time() #update time stamp return hermite_cache[(order, cache_key)][1] else: xmax, xa, ya, y2a=hermite_cache[order] y= spline.splint(xa, ya, y2a, x) if cache_key: hermite_cache[(order,cache_key)]=[time.time(), y] return y