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NR.py
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NR.py
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import math
import numpy as np
def dfridr(func,x,h,err):
CON = 1.4
CON2 = CON*CON
BIG = 1.e30
NTAB = 10
SAFE = 2.
if(h == 0.):
print 'h must be nonzero in dfridr'
hh = h
a = np.empty([NTAB,NTAB])
a[1,1] = (func(x+hh)-func(x-hh))/(2.0*hh)
err=BIG
for i in range(1,NTAB):
x = hh/CON
a[1,i] = (func(x+hh)-func(x-hh))/(2.0*hh)
fac = CON2
for j in range(1,i):
a[j,i] = (a[j-1,i]*fac-a[j-1,i-1])/(fac-1.)
fac = CON2*fac
errt = max(abs(a[j,i]-a[j-1,i]),abs(a[j,i]-a[j-1,i-1]))
if (errt <= err):
err = errt
dfridr = a[j,i]
if (abs(a[i,i]-a[i-1,i-1]) >= SAFE*err):
return
return
def rombint(f,a,b,tol):
# Rombint returns the integral from a to b of using Romberg integration.
# The method converges provided that f(x) is continuous in (a,b).
# f must be double precision and must be declared external in the calling
# routine. tol indicates the desired relative accuracy in the integral.
MAXITER = 40
MAXJ = 5
h = 0.5e0*(b-a)
gmax = h*(f(a)+f(b))
g = np.empty([MAXJ+1])
g[1] = gmax
nint = 1
error = 1.0e20
i = 1
while i > MAXITER or (i > 5 and abs(error) < tol) :
i = i+1 # Calculate next trapezoidal rule approximation to integral.
g0 = 0.0e0
for k in range(nint):
g0 = g0+f(a+(k+k-1)*h)
g0 = 0.5e0*g[1]+h*g0
h = 0.5e0*h
nint *= 2
jmax = min(i,MAXJ)
fourj = 1.0e0
for j in range(jmax): # Use Richardson extrapolation.
fourj = 4.0e0*fourj
g1 = g0+(g0-g[j])/(fourj-1.0e0)
g[j] = g0
g0 = g1
if (abs(g0) > tol):
error = 1.0e0-gmax/g0
else:
error = gmax
gmax = g0
g[jmax+1] = g0
else:
rombint = g0
if (i > MAXITER and abs(error) > tol):
print 'Rombint failed to converge; integral, error = ',rombint,' , ',error
return rombint
#########################################################################################################
#
# Fifth order Runge-Kutta Method with Adaptive Stepsize.
# Integrate 'func' with parameters array fp(np) which contains any extra parameters other than the
# integration variable from a to b, with initial step size dxinit and fractional accuracy eps.
#
# In other words,
# _b
# /
# |
# | FUNC(x,fp)dx
# |
# /
# _ a
#
# fp of length np, (i.e. real*8 fp(np)) contains all variables other
# than the integration variable, say, x.
#
#########################################################################################################
def INTEGRATE(FUNC,fp,np,a,b,dxinit,eps):
fp = np.empty([np])
maxsteps =1.e8
Nstep = 0
x = a
dx = dxinit
y = 0.e0
while (x-b)*(b-a) < 0.e0 and Nstep < maxsteps:
Nstep += 1
dydx = FUNC(x,fp,np)
yscale = max(abs(y) + abs(dx*dydx), 1.e-12)
if ((x+dx-b)*(x+dx-a) > 0.e0): dx = b - x # If stepsize overshoots, decrease it.
dxnext = RUNGE5VAR(y,dydx,x,dx,eps,yscale,dxnext,FUNC,fp,np)[2]
dx = dxnext
if (Nstep >= maxsteps): print 'WARNING: failed to converge in INTEGRATE.'
y = RUNGE5VAR(y,dydx,x,dx,eps,yscale,dxnext,FUNC,fp,np)[0]
INTEGRATE = y
return
###################################################################################################################
#
# Fifth-order Runge-Kutta step with monitoring of local truncation error to ensure accuracy and adjust stepsize.
# Input are the dependent-variable y and its derivative dydx at the starting value of the independent variable x.
# Also input are the stepsize to be attempted htry, the required accuracy eps, and the value yscale, against
# which the error is scaled. On output, y and x are replaced by their new values. hdid is the stepsize that
# was actually accomplished, and hnext is the estimated next stepsize. DERIVS is the user-supplied routine that
# computes right-hand-side derivatives. The argument fparm is for an optional second argument to DERIVS
# (NOT integrated over).
#
###################################################################################################################
def RUNGE5VAR(y,dydx,x,htry,eps,yscale,hnext,DERIVS,fp,np):
safety = 0.9e0
pgrow = -0.2e0
pshrink = -0.25e0
errcon = 1.89e-4
errmax = 10.e0
ytemp = 0.
yerr = 0.
h = htry # Set stepsize to initial accuracy.
while errmax > 1.e0 :
yerr = RUNGE(y,dydx,x,h,ytemp,yerr,DERIVS,fp,np)[1]
errmax = abs(yerr/yscale)/eps # Scale relative to required accuracy.
if (errmax > 1.e0): # Truncation error too large; reduce h
htemp = safety*h*(errmax**pshrink)
hold = h
h = math.copysign(max(abs(htemp),0.1e0*abs(h)),h) # No more than factor of 10
xnew = x + h
if (xnew == x):
print 'WARNING: ','Stepsize underflow in RUNGE5VAR().'
h = hold
errmax = 0.e0
# Step succeeded. Compute estimated size of next step.
if (errmax > errcon):
hnext = safety*h*(errmax**pgrow)
else:
hnext = 5.e0 * h # No more than factor of 5 increase.
x = x + h
y = ytemp
return y,x,hnext
###################################################################################################################
#
# Given values for a variable y and its derivative dydx known at x, use the fifth-order Cash-Karp Runge-Kutta
# method to advance the solution over an interval h and return the incremented variables as yout. Also
# return an estimate of the local truncation error in yout using the embedded fourth order method. The user
# supplies the routine DERIVS(x,y,dydx), which returns derivatives dydx at x.
#
###################################################################################################################
def RUNGE(y,dydx,x,h,yout,yerr,DERIVS,fp,np):
a2 = 0.2e0
a3 = 0.3e0
a4 = 0.6e0
a5 = 1.e0
a6 = 0.875e0
c1 = 37.e0/378.e0
c3 = 250.e0/621.e0
c4 = 125.e0/594.e0
c6 = 512.e0/1771.e0
dc1 = c1 - 2825.e0/27648.e0
dc3 = c3 - 18575.e0/48384.e0
dc4 = c4 - 13525.e0/55296.e0
dc5 = -277.e0/14336.e0
dc6 = c6 - 0.25e0
ak3 = DERIVS(x+a3*h,fp,np)
ak4 = DERIVS(x+a4*h,fp,np)
ak5 = DERIVS(x+a5*h,fp,np)
ak6 = DERIVS(x+a6*h,fp,np)
# Estimate the fifth order value.
yout = y + h*(c1*dydx + c3*ak3 + c4*ak4 + c6*ak6)
# Estimate error as difference between fourth and fifth order
yerr = h*(dc1*dydx + dc3*ak3 + dc4*ak4 + dc5*ak5 + dc6*ak6)
return yout,yerr
###################################################################################################################
# Spline fit subroutines
##########################
# returned: y2
def spline(x,y,n,yp1,ypn,y2):
NMAX=100010
u = np.empty([NMAX])
if (yp1 > 0.99e30):
y2[0] = 0.e0
u[0] = 0.e0
else:
y2[0] = -0.5e0
u[0] = (3.e0/(x[2]-x[1]))*((y[2]-y[1])/(x[2]-x[1])-yp1)
for i in range(1,n):
sig = (x[i]-x[i-1])/(x[i+1]-x[i-1])
p = sig*y2[i-1]+2.e0
y2[i] = (sig-1.e0)/p
u[i] = (6.e0*((y[i+1]-y[i])/(x[i+1]-x[i])-(y[i]-y[i-1])/(x[i]-x[i-1]))/(x[i+1]-x[i-1])-sig*u[i-1])/p
if (ypn > .99e30):
qn=0.e0
un=0.e0
else:
qn=0.5e0
un=(3.e0/(x[n]-x[n-1]))*(ypn-(y[n]-y[n-1])/(x[n]-x[n-1]))
y2[n]=(un-qn*u[n-1])/(qn*y2[n-1]+1.e0)
for k in range(n-1,0,-1):
y2[k]=y2[k]*y2[k+1]+u[k]
return y2
#---------------------------------------------------------------------------------
# returned: y
def splint(xa,ya,y2a,n,x,y):
klo = 1
khi = n
while khi-klo > 1 :
k=(khi+klo)/2
if(xa[k] > x):
khi=k
else:
klo=k
h=xa[khi]-xa[klo]
if(h == 0.):
print 'bad xa input in splint'
a = (xa[khi]-x)/h
b = (x-xa[klo])/h
y = a*ya[klo]+b*ya[khi]+((a**3-a)*y2a[klo]+(b**3-b)*y2a[khi])*(h**2)/6.
return y
#---------------------------------------------------------------------------------
# random numer generator2, returned: ran2
def ran2(idum):
IM1 = 2147483563
IM2 = 2147483399
AM = 1./IM1
IMM1 = IM1-1
IA1 = 40014
IA2 = 40692
IQ1 = 53668
IQ2 = 52774
IR1 = 12211
IR2 = 3791
NTAB = 32
NDIV = 1+IMM1/NTAB
EPS = 1.2e-7
RNMX = 1.-EPS
idum2 = 123456789
iv = np.zeros([NTAB])
iy = 0
if idum <= 0 :
idum = max(-idum,1)
idum2 = idum
for j in range(NTAB+8,0,-1):
k = idum/IQ1
idum = IA1*(idum-k*IQ1)-k*IR1
if (idum < 0): idum += IM1
if (j <= NTAB): iv[j] = idum
iy = iv[0]
k = idum/IQ1
idum = IA1*(idum-k*IQ1)-k*IR1
if (idum < 0): idum += IM1
k = idum2/IQ2
idum2 = IA2*(idum2-k*IQ2)-k*IR2
if (idum2 < 0): idum2 += IM2
j = 1+iy/NDIV
iy = iv[j]-idum2
iv[j] = idum
if(iy < 1): iy += IMM1
ran2 = min(AM*iy,RNMX)
return ran2
#---------------------------------------------------------------------------------
# returned: gasdev
def gasdev(idum):
iset = 0
rsq = 0
if (iset == 0) :
while rsq >= 1. or rsq == 0.:
v1 = 2.*ran1(idum)-1.
v2 = 2.*ran1(idum)-1.
rsq = v1**2+v2**2
fac = math.sqrt(-2.*math.log(rsq)/rsq)
gset = v1*fac
gasdev = v2*fac
iset = 1
else:
gasdev = gset
iset = 0
return gasdev
#---------------------------------------------------------------------------------
# random numer generator2, returned: ran1
def ran1(idum):
IA = 16807
IM = 2147483647
AM = 1./IM
IQ = 127773
IR = 2836
NTAB = 32
NDIV = 1+(IM-1)/NTAB
EPS = 1.2e-7
RNMX = 1.-EPS
iv = np.zeros([NTAB])
iy = 0
if idum <= 0 or iy == 0:
idum = max(-idum,1)
for j in range(NTAB+8,0,-1):
k = idum/IQ
idum = IA*(idum-k*IQ)-IR*k
if (idum < 0): idum += IM
if (j <= NTAB): iv[j] = idum
iy = iv[0]
k = idum/IQ
idum = IA*(idum-k*IQ)-IR*k
if (idum < 0): idum += IM
j = 1+iy/NDIV
iy = iv[j]
iv[j] = idum
ran1 = min(AM*iy,RNMX)
return ran1
#---------------------------------------------------------------------------------
# returned: a
def choldc(a,n,np,p):
for i in range(0,n):
for j in range(i,n):
sum = a[i,j]
for k in range(i-1,0,-1):
sum -= a[i,k]*a[j,k]
if(i == j):
if(sum <= 0.):
print 'choldc failed'
break
p[i] = math.sqrt(sum)
else:
a[j,i]=sum/p[i]
return a # CHECK RETURN VALUE
#---------------------------------------------------------------------------------
# returned: a,d,v
def jacobi(a,n,np,d,v,nrot):
NMAX = 500
b = np.empty([NMAX])
z = np.empty([NMAX])
for ip in range(n):
for iq in range(n):
v[ip,iq] = 0.
v[ip,ip] = 1.
for ip in range(n):
b[ip] = a[ip,ip]
d[ip] = b[ip]
z[ip] = 0.
nrot = 0
for i in range(50):
sm=0.
for ip in range(n-1):
for iq in range(ip+1,n):
sm += abs(a[ip,i])
if(sm == 0.): return a,d,v
if(i < 4):
tresh = 0.2*sm/n**2
else:
tresh = 0.
for ip in range(n-1):
for iq in range(ip+1,n):
g = 100.*abs(a[ip,iq])
if (i > 4) and (abs(d[ip])+g == abs(d(ip))) and (abs(d[iq])+g == abs(d[iq])):
a[p,iq]=0.
elif abs(a[ip,iq]) > tresh:
h=d[iq]-d[ip]
if(abs[h]+g == abs[h]):
t = a[ip,iq]/h
else:
theta = 0.5*h/a[ip,iq]
t = 1./(abs(theta)+math.sqrt(1.+theta**2))
if theta <0.: t = -t
c = 1./math.sqrt(1+t**2)
s = t*c
tau = s/(1.+c)
h = t*a[ip,iq]
z[ip] -= h
z[iq] += h
d[ip] -= h
d[iq] += h
a[ip,iq] = 0.
for j in range(ip-1):
g = a[j,ip]
h = a[j,iq]
a[j,ip] = g-s*(h+g*tau)
a[j,iq] = h+s*(g-h*tau)
for j in range(ip,iq-1):
g = a[ip,j]
h = a[j,iq]
a[ip,j] = g-s*(h+g*tau)
a[j,iq] = h+s*(g-h*tau)
for j in range(iq,n):
g = a[ip,j]
h = a[iq,j]
a[ip,j] = g-s*(h+g*tau)
a[iq,j] = h+s*(g-h*tau)
for j in range(n):
g = v[j,ip]
h = v[j,iq]
v[j,ip] = g-s*(h+g*tau)
v[j,iq] = h+s*(g-h*tau)
nrot += 1
for ip in range(n):
b[ip] += z[ip]
d[ip] = b[ip]
z[ip] = 0.
print 'too many iterations in jacobi'
return a,d,v # CHECK RETURN VALUE
#---------------------------------------------------------------------------------
# returned: ave,adev,sdev,var,skew,curt (moments of data1)
def moment(data1,n,ave,adev,sdev,var,skew,curt):
if(n <= 1):
print 'n must be at least 2 in moment'
s = 0.
for j in range(n):
s += data1[j]
ave = s/n
adev = 0.
var = 0.
skew = 0.
curt = 0.
eps = 0.
for j in range(n):
s = data1[j]-ave
ep += s
adev += abs(s)
p = s*s
var += p
p *= s
skew += p
p *= s
curt += p
adev /= n
var = (var-ep**2/n)/(n-1)
sdev = math.sqrt(var)
if(var != 0.):
skew /= (n*sdev**3)
curt /= (n*var**2)-3.
else:
print 'no skew or kurtosis when zero variance in moment'
return ave,adev,sdev,var,skew,curt
#---------------------------------------------------------------------------------
# returned: gammp
def gammp(a,x):
if x < 0. or a <= 0.:
print 'bad arguments in gammp'
if x < a+1.:
gammp = gser(gamser,a,x,gln)
else:
gammp = 1.- gcf(gammcf,a,x,gln)
return gammp
#---------------------------------------------------------------------------------
# returned: gammcf
def gcf(gammcf,a,x,gln):
ITMAX = 100
EPS = 3.e-7
FPMIN = 1.e-30
gln = gammln(a)
b = x+1.-a
c = 1./FPMIN
d = 1./b
h = d
for i in range(ITMAX):
an = -i*(i-a)
b += 2.
d = an*d+b
if abs(d) < FPMIN : d = FPMIN
c = b+an/c
if abs(c) < FPMIN : c = FPMIN
d = 1./d
de = d*c
h *= de
if abs(de-1.) < EPS :
print 'a too large, ITMAX too small in gcf'
break
gammcf = math.exp(-x+a*math.log(x)-gln)*h
return gammcf
#---------------------------------------------------------------------------------
# returned: gamser
def gser(gamser,a,x,gln):
ITMAX = 100
EPS = 3.e-7
gln = gammln(a)
if x <= 0. :
if(x < 0.):
print 'x < 0 in gser'
gamser=0.
return gamser
ap = a
sum = 1./a
de = sum
for n in range(ITMAX):
ap += 1.
de *= x/ap
sum += de
if abs(de) < abs(sum)*EPS :
print 'a too large, ITMAX too small in gser'
break
gamser = sum*exp(-x+a*math.log(x)-gln)
return gamser
#---------------------------------------------------------------------------------
# returned: gammln
def gammln(xx):
cof = np.array([76.18009172947146e0,-86.50532032941677e0,24.01409824083091e0,\
-1.231739572450155e0,.1208650973866179e-2,-.5395239384953e-5])
stp = np.array([76.18009172947146e0,-86.50532032941677e0,24.01409824083091e0,\
-1.231739572450155e0,.1208650973866179e-2,-.5395239384953e-5,2.5066282746310005e0])
x = xx
y = x
tmp = x+5.5e0
tmp = (x+0.5e0)*math.log(tmp)-tmp
ser=1.000000000190015e0
for j in range(6):
y += 1.e0
ser += cof[j]/y
gammln = tmp+np.log(stp*ser/x)
return gammln
#---------------------------------------------------------------------------------
# returned: rtbis
def rtbis(func,x1,x2,xacc):
JMAX = 500
fmid = func(x2)
f = func(x1)
if f*fmid >= 0.e0 :
print 'root must be bracketed in rtbis'
if f < 0.e0 :
rtbis = x1
dx = x2-x1
else:
rtbis = x2
dx = x1-x2
for j in range(JMAX):
dx *= 0.5e0
xmid = rtbis+dx
fmid = func(xmid)
if fmid <= 0.e0 : rtbis = xmid
if abs(dx) < xacc or fmid == 0.e0 : return rtbis
print 'too many bisections in rtbis'
#---------------------------------------------------------------------------------
# returned:
def zbrac(func,x1,x2,succes):
FACTOR = 1.6e0
NTRY = 1000
if x1 == x2:
print 'you have to guess an initial range in zbrac'
f1 = func(x1)
f2 = func(x2)
succes = True
for j in range(NTRY):
if f1*f2 < 0.e0 : return f1,f2
if abs(f1) < abs(f2) :
x1 += FACTOR*(x1-x2)
f1 = func(x1)
else:
x2 += FACTOR*(x2-x1)
f2 = func(x2)
succes = False
return f1,f2 # CHECK RETURN VALUE
#---------------------------------------------------------------------------------
# returned:
def zbrak(fx,x1,x2,n,xb1,xb2,nb):
nbb = 0
x = x1
dx = (x2-x1)/n
fp = fx(x)
for i in range(n):
x += dx
fc = fx(x)
if fc*fp < 0.e0 :
nbb += 1
xb1[nbb] = x-dx
xb2[nbb] = x
if nbb == nb : break
fp=fc
nb=nbb
return xb1,xb2,nb # CHECK RETURN VALUE
#---------------------------------------------------------------------------------
# returned: zbrent
def zbrent(func,x1,x2,tol):
ITMAX = 100
EPS = 3.e-8
a = x1
b = x2
fa = func(a)
fb = func(b)
if (fa >0. and fb > 0.) or (fa < 0. and fb < 0.) :
print 'root must be bracketed for zbrent'
c = b
fc = fb
for ite in range(ITMAX):
if (fb > 0 and fc > 0.) or (fb < 0. and fc < 0.) :
c = a
fc = fa
d = b-a
e = d
if abs(fc) < abs(fb) :
a = b
b = c
c = a
fa = fb
fb = fc
fc = fa
tol1 = 2.*EPS*abs(b)+0.5*tol
xm = .5*(c-b)
if abs(xm) <= tol1 or fb == 0. :
zbrent = b
return zbrent
if abs(e) >= tol1 and abs(fa) > abs(fb) :
s = fb/fa
if a == c :
p = 2.*xm*s
q = 1.-s
else:
q = fa/fc
r = fb/fc
p = s*(2.*xm*q*(q-r)-(b-a)*(r-1.))
q = (q-1.)*(r-1.)*(s-1.)
if p > 0. : q += -1.
p = abs(p)
if(2.*p < min(3.*xm*q-abs(tol1*q),abs(e*q))):
e = d
d = p/q
else:
d = xm
e = d
else:
d = xm
e = d
a = b
fa = fb
if abs(d) > tol1 :
b += d
else:
b += math.copysign(tol1,xm)
fb = func(b)
print 'zbrent exceeding maximum iterations'
zbrent = b
return zbrent
#---------------------------------------------------------------------------------
# returned: arr
def sort(n,arr):
NSTACK = 50
istack = np.empty([NSTACK])
jstack = 0
M = 7
l = 1
ir = n
while ir-l < M:
for j in range(l,ir):
a = arr[j]
for i in range(j,l,-1):
if arr[i] < a :
arr[i+1] = a
break
arr[i+1] = arr[i]
i = l-1 # POTENTIAL SOURCE OF BUG
arr[i+1]=a
if jstack == 0 : return arr
ir = istack[jstack]
l = istack[jstack-1]
jstack -= 2
else:
k = (l+ir)/2.
temp = arr[k]
arr[k] = arr[l+1]
arr[l+1] = temp
if arr[l] > arr[ir] :
temp = arr[l]
arr[l] = arr[ir]
arr[ir] = temp
if arr[l+1] > arr[ir] :
temp = arr[l+1]
arr[l+1] = arr[ir]
arr[ir] = temp
if arr[l] > arr[l+1] :
temp = arr[l]
arr[l] = arr[l+1]
arr[l+1] = temp
i = l+1
j = ir
a = arr[l+1]
while arr[i] < a and j > i:
i = i+1
while (arr[j] > a): #
j = j-1 # POTENTIAL SOURCE OF BUGS
temp = arr[i] # (ll. 904-912 fortran)
arr[i] = arr[j] #
arr[j] = temp #
arr[l+1] = arr[j] #
arr[j] = a
jstack += 2
if jstack > NSTACK :
print 'NSTACK too small in sort'
if ir-i+1 >= j-l :
istack[jstack] = ir
istack[jstack-1] = i
ir = j-1
else:
istack[jstack] = j-1
istack[jstack-1] = l
l = i
#---------------------------------------------------------------------------------
# returned: ra,rb,rc
def sort3(n,ra,rb,rc,wksp,iwksp):
ra,iwksp = indexx(n,ra,iwksp)
for j in range(n):
wksp[j] = ra[j]
for j in range(n):
ra[j] = wksp[iwksp[j]]
for j in range(n):
wksp[j] = rb[j]
for j in range(n):
rb[j] = wksp[iwksp[j]]
for j in range(n):
wksp[j] = rc[j]
for j in range(n):
rc[j] = wksp[iwksp[j]]
return ra,rb,rc # CHECK RETURN VALUE
#---------------------------------------------------------------------------------
# returned: arr,indx
def indexx(n,arr,indx):
M=7
NSTACK=50
istack = np.empty([NSTACK])
for j in range(1,n):
indx[j]=j
jstack=0
l=1
ir=n
while (ir-l < M):
if (ir-l < M):
for j in range(l+1,ir):
indxt=indx[j]
a=arr[indxt]
for i in range(j-1,1,-1): #
while (arr[indx[i]] <= a) is False: # POTENTIAL SOURCE OF BUGS
indx[i+1]=indx[i] # (ll. 978-983 fortran)
i=0 #
indx[i+1]=indxt #
if jstack == 0 : return arr,indexx
ir = istack[jstack]
l = istack[jstack-1]
jstack -= 2
else:
k=(l+ir)/2.
itemp=indx[k]
indx[k]=indx[l+1]
indx[l+1]=itemp
if(arr[indx[l+1]] > arr[indx[ir]]):
itemp=indx(l+1)
indx[l+1]=indx[ir]
indx[ir]=itemp
if(arr[indx[l]] > arr[indx[ir]]):
itemp=indx[l]
indx[l]=indx[ir]
indx[ir]=itemp
if(arr[indx[l+1]] > arr[indx[l]]):
itemp=indx[l+1]
indx[l+1]=indx[l]
indx[l]=itemp
i=l+1
j=ir
indxt=indx[l]
a=arr[indxt]
while (arr[indx[i]] < a): #
i=i+1 #
while (arr[indx[j]] > a): #
j=j-1 # POTENTIAL SOURCE OF BUGS
while (j < i): # (ll. 1014-1018 fortran)
itemp=indx[i] #
indx[i]=indx[j] #
indx[j]=itemp #
indx[l]=indx[j] #
indx[j]=indxt
jstack+=2
if(jstack > NSTACK):
print 'NSTACK too small in indexx'
break
if(ir-i+1 >= j-l):
istack[jstack]=ir
istack[jstack-1]=i
ir=j-1
else:
istack[jstack]=j-1
istack[jstack-1]=l
l=i
#---------------------------------------------------------------------------------
# returned: s
def qtrap(func,a,b,s):
EPS = 1.e-6
JMAX = 20
olds = 0.
for j in range(JMAX):
s = trapzd(func,a,b,s,j)
if j > 5 :
if abs(s-olds) < EPS*abs(olds) or (s == 0. and olds == 0.) : return s
olds=s
print 'too many steps in qtrap'
#---------------------------------------------------------------------------------
# returned: s
def trapzd(func,a,b,s,n):
if n == 1:
s = 0.5*(b-a)*(func(a)+func(b))
else:
it = 2**(n-2)
tnm = it
de = (b-a)/tnm
x = a+0.5*de
sum = 0.
for j in range(it):
sum += func(x)
x += de
s = 0.5*(s+(b-a)*sum/tnm)
return s
#---------------------------------------------------------------------------------
# returned: erf (error function for x)
def erf(x):
if x < 0. :
erf = -gammp(.5e0,x**2)
else:
erf = gammp(.500,x**2)
return erf
#---------------------------------------------------------------------------------
# Linear equation solution by Gauss-Jordan elimination, equation
# (2.1.1) above. a(1:n,1:n) is an input matrix stored in an array
# of physical dimensions np by np. b(1:n,1:m) is an input matrix
# containing the m right-hand side vectors, stored in an array of
# physical dimensions np by mp. On output, a(1:n,1:n) is replaced
# by its matrix inverse, and b(1:n,1:m) is replaced by the
# corresponding set of solution vectors.
# Parameter: NMAX is the largest anticipated value of n.
# returned: a, b
def gaussj(a,n,np,b,m,mp):
NMAX=50
indxc = np.empty([NMAX])
indxr = np.empty([NMAX])
ipiv = np.empty([NMAX])
for j in range(n):
ipiv[j] = 0
for i in range(n):
big = 0.
for j in range(n):
if ipiv[j] != 1 :
for k in range(n):
if ipiv[k] == 0 :
if abs(a[j,k]) >= big :
big = abs(a[j,k])
irow = j
icol = k
ipiv[icol] += 1
# We now have the pivot element, so we interchange rows, if needed,
# to put the pivot element on the diagonal. The columns are not
# physically interchanged, only relabeled:
# indxc(i), the column of the ith pivot element, is the ith column
# that is reduced, while indxr(i) is the row in which that pivot
# element was originally located. If indxr(i) =indxc(i) there is an
# implied column interchange. With this form of bookkeeping, the
# solution b's will end up in the correct order, and the inverse
# matrix will be scrambled by columns.
if irow != icol :
for l in range(n):
dum = a[irow,l]
a[irow,l] = a[icol,l]
a[icol,l] = dum
for l in range(m):
dum = b[irow,l]
b[irow,l] = b[icol,l]
b[icol,l] = dum
indxr[i]=irow
indxc[i]=icol
if a[icol,icol] == 0. :
print 'singular matrix in gaussj'
break
pivinv = 1./a[icol,icol]
a[icol,icol]=1.
for l in range(n):
a[icol,l] *= pivinv
for l in range(m):
b[icol,l] *= pivinv
for ll in range(n): # Next, we reduce the rows...
if ll != icol : #...except for the pivot one, of course.
dum = a[ll,icol]
a[ll,icol] = 0.
for l in range(n):
a[ll,l] -= a[icol,l]*dum
for l in range(m):
b[ll,l] -= b[icol,l]*dum