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92 lines
2.5 KiB
92 lines
2.5 KiB
10 years ago
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#
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# wpylib.math.linalg.gram_schmidt module
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# Created: 20150401
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# Wirawan Purwanto
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#
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"""
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wpylib.math.linalg.gram_schmidt
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Provides Gram-Schmidt orthogonalization of a vector set (a 2-D array).
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We will use the modified algorithm that is more stable numerically.
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Reference: http://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process
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"""
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def modgs_classic(V):
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"""Gram-Schmidt orthonormalization for the column vectors in matrix V.
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Classic routine, all hand-written, no acceleration.
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This is the reference implementation.
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"""
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from numpy import array, vdot, empty, outer, sqrt, real
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V = array(V, copy=True)
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assert len(V.shape) == 2
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numcols = V.shape[1]
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N_orig = empty((numcols,), dtype=V.dtype)
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for i in xrange(numcols):
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Vi = V[:,i]
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N_orig[i] = real(vdot(Vi, Vi)) # always a real number
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# FIXME: below could blow up if the norm is exactly zero
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Vi /= sqrt(N_orig[i])
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Ui = Vi # now Ui has been orthonormalized
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for j in xrange(i+1, numcols):
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Vj = V[:,j]
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proj_Ui_Vj = vdot(Ui, Vj)
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Vj -= proj_Ui_Vj * Ui
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return V, N_orig.real
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def modgs_fast1(V):
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"""Gram-Schmidt orthonormalization for the column vectors in matrix V.
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Fast(er) routine, replaced inner loop with mat-vec and outer products.
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"""
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from numpy import array, dot, vdot, empty, outer, sqrt, real
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V = array(V, copy=True)
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assert len(V.shape) == 2
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numcols = V.shape[1]
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N_orig = empty((numcols,), dtype=V.dtype)
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for i in xrange(numcols):
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Vi = V[:,i]
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N_orig[i] = real(vdot(Vi, Vi)) # always a real number
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# FIXME: below could blow up if the norm is exactly zero
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Vi /= sqrt(N_orig[i])
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Ui = Vi # now Ui has been orthonormalized
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# Now Vi is normalized -> renamed to Ui
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if i+1 < numcols:
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Vjj = V[:,i+1:]
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proj_u_Vjj = dot(Ui.conj(), Vjj)
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Vjj -= outer(Ui, proj_u_Vjj)
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return V, N_orig.real
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def modgs_einsum(V):
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"""Gram-Schmidt orthonormalization for the column vectors in matrix V.
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Fast(er) routine, using einsum.
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"""
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from numpy import array, vdot, einsum, empty, sqrt, real
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V = array(V, copy=True)
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assert len(V.shape) == 2
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numcols = V.shape[1]
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N_orig = empty((numcols,), dtype=V.dtype)
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for i in xrange(numcols):
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Vi = V[:,i]
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N_orig[i] = real(vdot(Vi, Vi)) # always a real number
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# FIXME: below could blow up if the norm is exactly zero
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Vi /= sqrt(N_orig[i])
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Ui = Vi # now Ui has been orthonormalized
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# Now Vi is normalized -> renamed to Ui
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if i+1 < numcols:
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Vjj = V[:,i+1:]
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Vjj -= einsum('i,j,jk', Ui, Ui.conj(), Vjj)
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return V, N_orig.real
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modgs = modgs_fast1
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