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* Created array indexer for "upper diagonal" indexing mode.

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Apparently I was able to create the non-iterative version of UDdec (1-based
  indices).
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Wirawan Purwanto 14 years ago
parent 961a802326
commit 93a89d2606
1 changed files with 322 additions and 2 deletions
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  1. 324
      math/symmetrix-array-index.PY

324
math/symmetrix-array-index.PY
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@ -2,7 +2,7 @@
"""
Symmetric array layout (lower diagonal, Fortran column-major ordering):
i >= j
----> j
| (1,1)
@ -11,7 +11,7 @@ i | (2,1) (2,2)
: : :
(N,1) (N,2) (N,3) ... (N,N)
In linear form:
The linear index goes like this:
1
2 N+1
@ -71,3 +71,323 @@ def copy_over_array(N, arr_L):
print "%5d %5d %5d %5d %5d" % (ii,jj,iskip,jskip,ldiag_index+1)
rslt[i,j] = arr_L[ldiag_index]
return rslt
# These are the reference implementation:
def LD(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 python convention; thus 0 <= 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
jskip = (jj)*N - (jj-1)*(jj)//2 # for 0-based
return iskip + jskip
def LDdec(ij, N):
"""Back-translates linear index 0, 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(N):
if jskip + (N - j) > ij:
jj = j
ii = ij - jskip + j
return (ii,jj)
jskip += N - j
raise ValueError, "LDdec(ij=%d,N=%d): invalid index ij" % (ij,N)
# end reference implementation
def test_LD_enc_dec(N):
"""Simple test to check LD encoding and decoding correctness.
For python-style indexing (0 <= i < N, similarly for j)."""
for i in xrange(N):
for j in xrange(N):
ij = LD(i,j,N)
(ii,jj) = LDdec(ij,N)
print "%3d %3d | %6d | %3d %3d" % (i,j, ij, ii,jj)
def test_LD_enc_dec_diagonal(N):
"""Simple test to check LD encoding and decoding correctness.
For python-style indexing (0 <= i < N, similarly for j).
For diagonal element only
"""
from numpy import sqrt
LDsize = N * (N+1) / 2
for i in xrange(N):
j = i
ij = LD(i,j,N)
(ii,jj) = LDdec(ij,N)
jj2 = int( sqrt(((LDsize) - ij) * 2) )
print "%3d %3d | %6d ( %6d %6d ) | %3d %3d // %3d %3d" % (
i,j,
ij,
ij-LDsize, -2*(ij-LDsize),
ii,jj,
-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)."""
from numpy import sqrt
LDsize = N * (N+1) / 2
for j in xrange(N):
for i in xrange(j,N):
ij = LD(i,j,N)
(ii,jj) = LDdec(ij,N)
jj2 = ( sqrt(((LDsize) - ij) * 2) )
#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,
ii,jj,
jj2)
###############################################################################
"""
UPPER DIAGONAL IMPLEMENTATION
j >= i
Should be easier (?)
Symmetric array layout (lower diagonal, Fortran column-major ordering):
----> j
| (1,1) (1,2) (1,3) ... (1,N)
i | (2,2) (2,3) ... (2,N)
v (3,3) ... (3,N)
: :
(N,N)
The linear index goes like this:
1 2 4 ... N*(N+1)/2-4
3 5 ... N*(N+1)/2-3
6 ... N*(N+1)/2-2
: N*(N+1)/2-1
N*(N+1)/2
# For 1-based indices:
iskip = i - j (easy)
jskip = j * (j+1) / 2
In the large j limit, jskip is approximately 0.5*j**2 .
This means that the j can be 'guessed' by:
j_guess = int( sqrt(ij * 2) )
= int( sqrt(j * (j + 1) + (i - j)*2) )
= int( sqrt(j**2 + 2*i - j) )
Remember that i <= j, so j_guess ranges from (inclusive endpoints):
j_guess_min = int( sqrt(j**2 - j + 2) )
j_guess_max = int( sqrt(j**2 + j) )
Note: since
sqrt((j-1)**2) = sqrt(j**2 - 2j + 1)
sqrt((j+1)**2) = sqrt(j**2 + 2j + 1)
this means that, for all j > 0,
j_guess_min >= j-1
j_guess_max = j
So only those two j values matter.
Once we get the j, we can get the i value easily.
# For 0-based indices:
iskip = i - j (easy)
jskip = (j+1) * (j+2) / 2 - 1
"""
def UD(i,j,N):
"""Translates a lower-diagonal index (ii <= jj) to linear index
0, 1, 2, 3, ...
This follows python convention; thus 0 <= 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 = (j+1) * (j+2) // 2 - 1 # for 0-based
return iskip + jskip
def UDdec(ij, N):
"""Back-translates linear index 0, 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.
Derived from the 1-based version (UDdec1) below.
"""
jskip = 0
for j in xrange(1,N+1):
jskip = j * (j+1) // 2
if ij < jskip:
i = ij - jskip + j
return (i,j-1)
def UD1(i,j,N):
"""The 1-based version of UD
"""
# 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
jskip = (j) * (j+1) // 2 # for 1-based
return iskip + jskip
def UDdec1(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):
jskip = j * (j+1) // 2
if ij < jskip + 1:
"""
ij = iskip + jskip
-> iskip = ij - jskip = i - j
-> i = ij - jskip + j
"""
i = ij - jskip + j
return (i,j)
raise ValueError, "UDdec1(ij=%d,N=%d): invalid index ij" % (ij,N)
def UDdec1_v2(ij, N):
"""Back-translates linear index 1, 2, 3, ... to a lower-diagonal
index pair (ii <= jj).
Version 2, avoiding loop at a cost of doing sqrt() call.
"""
from numpy import sqrt
j = int(sqrt(ij*2+1))
jskip = j * (j+1) // 2
if ij < jskip + 1:
pass # correct already
else:
j = j + 1
jskip = j * (j+1) // 2
i = ij - jskip + j
return (i,j)
def UDdec1_v3(ij, N):
"""Back-translates linear index 1, 2, 3, ... to a lower-diagonal
index pair (ii <= jj).
Version 3, avoiding loop at a cost of doing sqrt() call.
"""
from numpy import sqrt
j = int(sqrt(ij*2)) # +1 not needed?
jskip = j * (j+1) // 2
if ij < jskip + 1:
pass # correct already
else:
j = j + 1
jskip = j * (j+1) // 2
i = ij - jskip + j
return (i,j)
def test_UD_enc_dec(N):
"""Simple test to check UD encoding and decoding correctness.
For python-style indexing (0 <= i < N, similarly for j)."""
# Success for N = 5, 8
for j in xrange(N):
for i in xrange(0,j+1):
ij = UD(i,j,N)
(ii,jj) = UDdec(ij,N)
# print "%3d %3d | %6d | %3d %3d" % (i,j, ij, ii,jj)
print "%3d %3d | %6d | %3d %3d >> %1s" % (i,j, ij, ii,jj, ("v" if i==ii and j==jj else "X"))
def test_UD_enc_dec1(N):
"""Simple test to check UD encoding and decoding correctness.
For python-style indexing (0 <= i < N, similarly for j)."""
for j in xrange(1,N+1):
for i in xrange(1,j+1):
ij = UD1(i,j,N)
(ii,jj) = UDdec1(ij,N)
print "%3d %3d | %6d | %3d %3d >> %1s" % (i,j, ij, ii,jj, ("v" if i==ii and j==jj else "X"))
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)."""
from numpy import sqrt
ok = True
for j in xrange(1,N+1):
for i in xrange(1,j+1):
ij = UD1(i,j,N)
(ii,jj) = UDdec1(ij,N)
#iii = i
#jjj = int(sqrt(ij*2+1))
#(iii,jjj) = UDdec1_v2(ij,N) # -- tested OK empirically till N=1000
(iii,jjj) = UDdec1_v3(ij,N) # also tested OK empirically till N=1000
if not (i==iii and j==jjj):
ok=False
print "%3d %3d | %6d %6d | %3d %3d : %3d %3d >> %1s" % (
i,j,
ij, ij*2,
ii,jj,
iii, jjj,
("v" if i==iii and j==jjj else "X"))
print ("ok" if ok else "NOT OK")

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