# [Numpy-svn] r3323 - in trunk/numpy: lib linalg

numpy-svn at scipy.org numpy-svn at scipy.org
Fri Oct 13 12:08:26 CDT 2006

```Author: charris
Date: 2006-10-13 12:08:24 -0500 (Fri, 13 Oct 2006)
New Revision: 3323

Modified:
trunk/numpy/lib/polynomial.py
trunk/numpy/linalg/linalg.py
Log:

Add a rcond parameter to the polyfit function and give it the double precision
default value that dgelsd uses (rcondx=-1) on the principle of least surprise.
Values of rcond less than this can also be useful, so a warning message and a
bit of explanation was added to the documentation.

The default value used by lstsq was set to the default (-1), and rcond in pinv
has a default value of 1e-15.

Modified: trunk/numpy/lib/polynomial.py
===================================================================
--- trunk/numpy/lib/polynomial.py	2006-10-13 11:52:25 UTC (rev 3322)
+++ trunk/numpy/lib/polynomial.py	2006-10-13 17:08:24 UTC (rev 3323)
@@ -30,13 +30,13 @@
get_linalg_funcs()
return eigvals(arg)

-def _lstsq(X, y):
+def _lstsq(X, y, rcond):
"Do least squares on the arguments"
try:
-        return lstsq(X, y)
+        return lstsq(X, y, rcond)
except TypeError:
get_linalg_funcs()
-        return lstsq(X, y)
+        return lstsq(X, y, rcond)

def poly(seq_of_zeros):
""" Return a sequence representing a polynomial given a sequence of roots.
@@ -169,10 +169,10 @@
val = poly1d(val)
return val

-def polyfit(x, y, N):
+def polyfit(x, y, N, rcond=-1):
"""

-    Do a best fit polynomial of order N of y to x.  Return value is a
+    Do a best fit polynomial of degree N of y to x.  Return value is a
vector of polynomial coefficients [pk ... p1 p0].  Eg, for N=2

p2*x0^2 +  p1*x0 + p0 = y1
@@ -195,8 +195,23 @@

p = (XT*X)^-1 * XT * y

-    where XT is the transpose of X and -1 denotes the inverse.
+    where XT is the transpose of X gand -1 denotes the inverse. However, this
+    method is susceptible to rounding errors and generally the singular value
+    decomposition is preferred and that is the method used here. The singular
+    value method takes a paramenter, 'rcond', which sets a limit on the
+    relative size of the smallest singular value to be used in solving the
+    equation. The default value of rcond is the double precision machine
+    precision as the actual solution is carried out in double precision.

+    WARNING: Power series fits are full of pitfalls for the unwary once the
+    degree of the fit get up around 4 or 5. Computation of the polynomial
+    values are sensitive to coefficient errors and the Vandermonde matrix is
+    ill conditioned. The knobs available to tune the fit are degree and rcond.
+    The rcond knob is a bit flaky and it can be useful to use values of rcond
+    less than the machine precision, 1e-24 for instance, but the quality of the
+    resulting fit needs to be checked against the data. The quality of
+    polynomial fits *can not* be taken for granted.
+
http://mathworld.wolfram.com/LeastSquaresFittingPolynomial.html,
but note that the k's and n's in the superscripts and subscripts
@@ -205,12 +220,10 @@

"""
-    x = NX.asarray(x)+0.
-    y = NX.asarray(y)+0.
-    y = NX.reshape(y, (len(y), 1))
-    X = vander(x, N+1)
-    c, resids, rank, s = _lstsq(X, y)
-    c.shape = (N+1,)
+    x = NX.asarray(x) + 0.0
+    y = NX.asarray(y) + 0.0
+    v = vander(x, N+1)
+    c, resids, rank, s = _lstsq(v, y, rcond)
return c

Modified: trunk/numpy/linalg/linalg.py
===================================================================
--- trunk/numpy/linalg/linalg.py	2006-10-13 11:52:25 UTC (rev 3322)
+++ trunk/numpy/linalg/linalg.py	2006-10-13 17:08:24 UTC (rev 3323)
@@ -140,12 +140,12 @@
allaxes.remove(k)
allaxes.insert(an, k)
a = a.transpose(allaxes)
-
+
oldshape = a.shape[-(an-b.ndim):]
prod = 1
for k in oldshape:
prod *= k
-
+
a = a.reshape(-1,prod)
b = b.ravel()
res = solve(a, b)
@@ -184,17 +184,17 @@
def invtensor(a, ind=2):
"""Find the inverse tensor.

-    ind > 0 ==> first (ind) indices of a are summed over
+    ind > 0 ==> first (ind) indices of a are summed over
ind < 0 ==> last (-ind) indices of a are summed over

if ind is a list, then it specifies the summed over axes

When the inv tensor and the tensor are summed over the
-    indicated axes a separable identity tensor remains.
+    indicated axes a separable identity tensor remains.

The inverse has the summed over axes at the end.
"""
-
+
a = asarray(a)
oldshape = a.shape
prod = 1
@@ -217,8 +217,8 @@
a = a.reshape(-1,prod)
ia = inv(a)
return ia.reshape(*invshape)
-

+
# Matrix inversion

def inv(a):
@@ -250,7 +250,7 @@

def qr(a, mode='full'):
"""cacluates A=QR, Q orthonormal, R upper triangular
-
+
mode:  'full' --> (Q,R)
'r'    --> R
'economic' --> A2 where the diagonal + upper triangle
@@ -267,8 +267,8 @@
routine_name = 'zgeqrf'
else:
lapack_routine = lapack_lite.dgeqrf
-        routine_name = 'dgeqrf'
-
+        routine_name = 'dgeqrf'
+
# calculate optimal size of work data 'work'
lwork = 1
work = zeros((lwork,), t)
@@ -307,7 +307,7 @@

if isComplexType(t):
lapack_routine = lapack_lite.zungqr
-        routine_name = 'zungqr'
+        routine_name = 'zungqr'
else:
lapack_routine = lapack_lite.dorgqr
routine_name = 'dorgqr'
@@ -326,7 +326,7 @@

if results['info'] > 0:
raise LinAlgError, '%s returns %d' % (routine_name, results['info'])
-
+
q = a[:mn,:].transpose()

if (q.dtype != result_t):
@@ -476,7 +476,7 @@
v[ind[2*i]] = vr[ind[2*i]] + 1j*vr[ind[2*i+1]]
v[ind[2*i+1]] = vr[ind[2*i]] - 1j*vr[ind[2*i+1]]
result_t = _complexType(result_t)
-
+
if results['info'] > 0:
raise LinAlgError, 'Eigenvalues did not converge'
vt = v.transpose().astype(result_t)
@@ -534,7 +534,7 @@
u,s,vh = svd(a)

If a is an M x N array, then the svd produces a factoring of the array
-    into two unitary (orthogonal) 2-d arrays u (MxM) and vh (NxN) and a
+    into two unitary (orthogonal) 2-d arrays u (MxM) and vh (NxN) and a
min(M,N)-length array of singular values such that

a == dot(u,dot(S,vh))
@@ -613,10 +613,10 @@

# Generalized inverse

-def pinv(a, rcond = 1.e-10):
+def pinv(a, rcond=1e-15 ):
"""Return the (Moore-Penrose) pseudo-inverse of a 2-d array

-    This method computes the generalized inverse using the
+    This method computes the generalized inverse using the
singular-value decomposition and all singular values larger than
rcond of the largest.
"""
@@ -660,18 +660,18 @@

# Linear Least Squares

-def lstsq(a, b, rcond=1.e-10):
+def lstsq(a, b, rcond=-1):
"""returns x,resids,rank,s
-where x minimizes 2-norm(|b - Ax|)
-      resids is the sum square residuals
-      rank is the rank of A
-      s is the rank of the singular values of A in descending order
+    where x minimizes 2-norm(|b - Ax|)
+    resids is the sum square residuals
+    rank is the rank of A
+    s is the rank of the singular values of A in descending order

-If b is a matrix then x is also a matrix with corresponding columns.
-If the rank of A is less than the number of columns of A or greater than
-the number of rows, then residuals will be returned as an empty array
-otherwise resids = sum((b-dot(A,x)**2).
-Singular values less than s[0]*rcond are treated as zero.
+    If b is a matrix then x is also a matrix with corresponding columns.
+    If the rank of A is less than the number of columns of A or greater than
+    the number of rows, then residuals will be returned as an empty array
+    otherwise resids = sum((b-dot(A,x)**2).
+    Singular values less than s[0]*rcond are treated as zero.
"""
import math
a = asarray(a)

```