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* Added faster LDdec1 for 1-based indexing.

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Tested through N=1000.
This commit is contained in:
Wirawan Purwanto
2012-01-26 10:46:55 -05:00
parent d04ea8cbea
commit 050e2414ef
1 changed files with 121 additions and 23 deletions
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144
math/symmetrix-array-index.PY
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@@ -43,7 +43,10 @@ Note:
"""
"""
Update 20120124: the jskip formula can be written in similar fashion to
Update 20120124:
Faster LDdec is possible.
The jskip formula can be written in similar fashion to
the 'UD' array format, as shown below.
The endpoint of the array index is N(N+1)/2 .
@@ -59,9 +62,36 @@ N(N+1)/2 - (N+1-j)(N+2-j) / 2 =
= (j-1)N + (j-2)(j-1)/2
>>> the same formula as before.
We can now make a similar analysis as in UD case to make a j_guess
formula:
j_guess = int( N+1 - sqrt((N(N+1)/2 - ij) * 2) )
Note that:
ij = N(N+1)/2 - (N+1-j)(N+2-j)/2 + i-j+1
Now focus on this expression:
xj := ( N(N+1)/2 - ij) * 2
= (N+1-j)*(N+2-j) - 2*(i+1-j)
So the maximum value of xj (for i=j) is:
xj_max = (N+1-j)*(N+2-j) - 2
= (N+1-j)**2 + (N+1-j) - 2
xj_min = (N+1-j)*(N+2-j) - 2*(N+1-j)
= (N+1-j)**2 - (N+1-j)
Again, these values satisfy the inequality
(N-j)**2 < xj_min <= xj_max < (N+2-j)**2
Thus translates to
N-j <= int(sqrt(xj)) <= N+1-j
or
j <= j_guess <= j+1
"""
import numpy
@@ -115,10 +145,30 @@ def LD(i,j,N):
jj = i
iskip = ii - jj # + 1
#jskip = (jj-1)*N - (jj-2)*(jj-1)/2 # for 1-based
jskip = (jj)*N - (jj-1)*(jj)//2 # for 0-based
return iskip + jskip
def LD1(i,j,N):
"""python equivalent of gafqmc_LD on nwchem-gafqmc integral
dumper module.
Translates a lower-diagonal index (ii >= jj) to linear index
0, 1, 2, 3, ...
This follows Fortran convention; thus 1 <= i <= N, and so also j.
"""
# iskip is row traversal, jskip is column traversal.
# (iskip+jskip) is the final array index.
if i >= j:
ii = i
jj = j
else:
ii = j
jj = i
iskip = ii - jj + 1
jskip = (jj-1)*N - (jj-2)*(jj-1)//2 # for 1-based
return iskip + jskip
def LDdec(ij, N):
"""Back-translates linear index 0, 1, 2, 3, ... to a lower-diagonal
@@ -137,6 +187,38 @@ def LDdec(ij, N):
raise ValueError, "LDdec(ij=%d,N=%d): invalid index ij" % (ij,N)
def LDdec1(ij, N):
"""Back-translates linear index 1, 2, 3, ... to a lower-diagonal
index pair (ii >= jj).
This is not optimal, but it avoids storing an auxiliary array
that is easily computable. Plus, this function is supposed to
be called rarely.
"""
jskip = 0
for j in xrange(1, N+1):
if jskip + (N + 1 - j) >= ij:
jj = j
ii = ij - jskip + j - 1
return (ii,jj)
jskip += (N + 1 - j)
raise ValueError, "LDdec1(ij=%d,N=%d): invalid index ij" % (ij,N)
def LDdec1_v2(ij, N):
"""Version 2, avoiding loop, but adding sqrt() function
"""
from numpy import sqrt
LDsize = N*(N+1) // 2
j = N + 1 - int( sqrt((LDsize - ij) * 2) )
jskip = (j-1)*N - (j-2)*(j-1)//2
if ij > jskip:
pass # correct already
else:
j = j - 1
jskip = (j-1)*N - (j-2)*(j-1)//2
i = ij - jskip + j - 1
return (i,j)
# end reference implementation
def test_LD_enc_dec(N):
@@ -169,21 +251,6 @@ def test_LD_enc_dec_diagonal(N):
-1, jj2)
# ^^ distance from end of array
"""
Faster LDdec is possible.
Consider:
jskip = (jj)*N - (jj-1)*(jj)/2 # for 0-based
= jj*(2*N - (jj-1)) / 2
"""
def Hack2_LD_enc_dec(N):
"""Simple test to check LD encoding and decoding correctness.
For python-style indexing (0 <= i < N, similarly for j)."""
@@ -194,12 +261,43 @@ def Hack2_LD_enc_dec(N):
ij = LD(i,j,N)
(ii,jj) = LDdec(ij,N)
jj2 = ( sqrt(((LDsize) - ij) * 2) )
j_guess = int(N + 1 - jj2) # for some reason this is the one that works for 0-based index
ok1 = (jj <= j_guess)
ok2 = (j_guess <= jj+1)
ok = ((jj <= j_guess) and (j_guess <= jj+1))
#print "%3d %3d | %6d | %3d %3d" % (i,j, ij, ii,jj)
print "%3d %3d | %6d %6d | %3d %3d // %8.4f" % (
i,j,
ij, (LDsize-ij) * 2,
ii,jj,
jj2)
if not ok:
# Verified OK empirically till N=1000.
print "%3d %3d | %6d %6d | %3d %3d // %8.4f %3d %c %d %d" % (
i,j,
ij, (LDsize-ij) * 2,
ii,jj,
jj2, j_guess, ("." if ok else "X"), ok1,ok2)
def Hack3_LD_enc_dec(N, print_all=False):
"""Simple test to check LD encoding and decoding correctness.
For Fortran-style indexing (1 <= i <= N, similarly for j)."""
from numpy import sqrt
LDsize = N * (N+1) / 2
for j in xrange(1,N+1):
for i in xrange(j,N+1):
ij = LD1(i,j,N)
(ii,jj) = LDdec1(ij,N)
(ii,jj) = LDdec1_v2(ij,N)
jj2 = ( sqrt(((LDsize) - ij) * 2) )
j_guess = N + 1 - int(jj2)
OK = (ii==i and jj==j)
ok1 = (jj <= j_guess)
ok2 = (j_guess <= jj+1)
ok = ((jj <= j_guess) and (j_guess <= jj+1))
#print "%3d %3d | %6d | %3d %3d" % (i,j, ij, ii,jj)
if print_all or not (OK and ok):
# Verified OK empirically till N=1000.
print "%3d %3d | %6d %6d | %3d %3d %c // %8.4f %3d %c %d %d" % (
i,j,
ij, (LDsize-ij) * 2,
ii,jj, ("." if OK else "X"),
jj2, j_guess, ("." if ok else "X"), ok1,ok2)
@@ -391,7 +489,7 @@ def test_UD_enc_dec1(N):
def hack1_UD_enc_dec1(N):
"""Simple test to check UD encoding and decoding correctness.
For python-style indexing (0 <= i < N, similarly for j)."""
For Fortran-style indexing (1 <= i <= N, similarly for j)."""
from numpy import sqrt
ok = True
for j in xrange(1,N+1):