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