def g(x):
"""
Integrates to 1000 over [0,10].
"""
return 3*x**2
def h(x):
"""
Integrates to 1.e11 + 1.e9 + 1000 over [0,10].
"""
return 11*x**10 + x**9 + 3*x**2
print 'Single order-10 CC & F1 quads for f=1, g=3*x**2, h(x)=10th degree:'
print '(Exact results: 10 1000 101000001000)'
cc10 = ClenshawCurtis(10, 0, 10)
print cc10.quad(f), cc10.quad(g), cc10.quad(h)
f10 = Fejer1(10, 0, 10)
print f10.quad(f), f10.quad(g), f10.quad(h)
print 'Composite quad for above cases:'
print cq.quad(f), cq.quad(g), cq.quad(h)
print
def plm3(x):
"""
Power-law integrand; integral = -1/x**2
"""
return 2 * x**(-3.)
def plm2(x):
"""
Power-law integrand; integral = -1/x.
Rolling exponential, -.6 at 1, -.4 at 10
"""
return exp((alpha + .2*(x-.5*range)/range)*x)
def rexp_map(y, alpha=alpha, range=range):
"""
Rolling exponential, transformed.
"""
x = x_y(y)
return exp((alpha + .2*(x-.5*range)/range)*x) / exp(alpha*x)
l, u = 1., range
y_l, y_u = y_x(l), y_x(u)
for n in [5, 10, 15, 25, 50, 100, 200, 500]:
ccx = ClenshawCurtis(n, l, u)
ccy = ClenshawCurtis(n, y_l, y_u)
xq = ccx.quad(rexp)
yq = ccy.quad(rexp_map)
print n, xq, yq
semilogy(ccx.nodes, rexp(ccx.nodes), 'b.')
xlabel('$x$')
ylabel('$f(x)$')
figure()
semilogy(ccy.nodes, rexp_map(ccy.nodes), 'g.')
cc = ClenshawCurtis(25, y_l, y_u)
semilogy(cc.nodes, rexp_map(cc.nodes), 'r.')
xlabel('$y$')
ylabel(r'$f(x)/\exp(\alpha x)$')
show()
def g(x):
"""
Integrates to 1000 over [0,10].
"""
return 3 * x**2
def h(x):
"""
Integrates to 1.e11 + 1.e9 + 1000 over [0,10].
"""
return 11 * x**10 + x**9 + 3 * x**2
print 'Single order-10 CC & F1 quads for f=1, g=3*x**2, h(x)=10th degree:'
print '(Exact results: 10 1000 101000001000)'
cc10 = ClenshawCurtis(10, 0, 10)
print cc10.quad(f), cc10.quad(g), cc10.quad(h)
f10 = Fejer1(10, 0, 10)
print f10.quad(f), f10.quad(g), f10.quad(h)
print 'Composite quad for above cases:'
print cq.quad(f), cq.quad(g), cq.quad(h)
print
def plm3(x):
"""
Power-law integrand; integral = -1/x**2
"""
return 2 * x**(-3.)
def plm2(x):
"""
Power-law integrand; integral = -1/x.
pl1 = -beta + .5
return (x/mid)**pl1 / (1. + x/mid)
def roll_map(y, beta=beta, range=range, mid=mid):
"""
Rolling exponential, transformed.
"""
x = x_y(y)
return roll(x) * x**beta
l, u = 1., range
y_l, y_u = y_x(l), y_x(u)
for n in [5, 10, 15, 25, 50, 100, 200, 500]:
ccx = ClenshawCurtis(n, l, u)
ccy = ClenshawCurtis(n, y_l, y_u)
xq = ccx.quad(roll)
yq = ccy.quad(roll_map)
print n, xq, yq
loglog(ccx.nodes, roll(ccx.nodes), 'b.')
cc = ClenshawCurtis(25, l, u)
loglog(cc.nodes, roll(cc.nodes), 'r.')
xlabel('$x$')
ylabel('$f(x)$')
figure()
loglog(-ccy.nodes, roll_map(ccy.nodes), 'g.')
cc = ClenshawCurtis(25, y_l, y_u)
loglog(-cc.nodes, roll_map(cc.nodes), 'r.')
xlabel('$-y$')
ylabel(r'$f(x)/x^{-\beta}$')
show()
def roll_map(y, beta=beta, range=range, mid=mid):
"""
Rolling exponential, transformed.
"""
x = x_y(y)
return roll(x) * x**beta
l, u = 1., range
y_l, y_u = y_x(l), y_x(u)
for n in [5, 10, 15, 25, 50, 100, 200, 500]:
ccx = ClenshawCurtis(n, l, u)
ccy = ClenshawCurtis(n, y_l, y_u)
xq = ccx.quad(roll)
yq = ccy.quad(roll_map)
print n, xq, yq
loglog(ccx.nodes, roll(ccx.nodes), 'b.')
cc = ClenshawCurtis(25, l, u)
loglog(cc.nodes, roll(cc.nodes), 'r.')
xlabel('$x$')
ylabel('$f(x)$')
figure()
loglog(-ccy.nodes, roll_map(ccy.nodes), 'g.')
cc = ClenshawCurtis(25, y_l, y_u)
loglog(-cc.nodes, roll_map(cc.nodes), 'r.')
xlabel('$-y$')
ylabel(r'$f(x)/x^{-\beta}$')
show()
