unsymmetric eigenvalue update

Patrick Alken patrick.alken@colorado.edu
Wed Jun 14 16:31:00 GMT 2006


Hi all,

  After reviewing the LAPACK and EISPACK eigenvalue routines,
I've changed my convergence criteria to declare failure after
30*N iterations. My original number of 30 was based on the
Numerical Recipes algorithm, which I now see was copied
from the EISPACK algorithm but with a typo of 30 instead of
30*N.

  I have run my code through Brian Gough's test program
on 12x12 and 200x200 random matrices, as well as some integer
matrix cases and find no significant disagreement with LAPACK.

  For those who expressed concern earlier about speed issues,
I'd like to share some things I found while studying this
problem:

1) In 1962 the Francis double-shift QR algorithm was published
   which typically converges in about 8n^3 flops if you only
   want eigenvalues. This is the algorithm which was implemented
   in EISPACK (hqr.f) and is also implemented in LAPACK as
   DLAHQR. Also this is the algorithm discussed in Golub &
   Van Loan and is presented in Numerical Recipes (which is just
   the EISPACK hqr routine converted to C).

2) In 1989, Bai and Demmel published their multi-shift QR
   algorithm and gave some numerical tests showing that it
   is generally faster than hqr from EISPACK. However they
   say its too complicated to derive an approximate number of
   flops required for the general problem. Furthermore,
   the multi-shift algorithm needs to call the double-shift
   algorithm to calculate the shifts. So it is impossible
   to dispense with the double-shift algorithm. This algorithm
   is implemented in LAPACK as DGEES (DGEES calls DLAHQR to
   get the shifts)

3) In a further significant development, in 2001 a multishift
   QR algorithm was proposed in the papers

   K. Braman et al, "The Multishift QR Algorithm: Part I:
   Maintaining Well Focused Shifts"

   and

   K. Braman et al, "The Multishift QR Algorithm: Part II:
   Aggressive Early Deflation"

   which introduces a way to deflate eigenvalues much more
   quickly than the standard test used in the previous two
   methods. The numerical results in these papers are very
   impressive and show convergence with far less flops than
   the LAPACK algorithm. Subsequent papers have extended this
   method to the QZ algorithm of the generalized eigenvalue
   problem, and as far as I can tell, this aggressive deflation
   method is considered the best QR algorithm currently
   available. If any experts are reading this please correct
   me!

So, the code in my gsl patch is equivalent to the HQR routine
in EISPACK, which has perfectly fine accuracy but is not
the fastest converging. I believe ultimately, the goal should
be to implement method #3 which will perform better than
LAPACK's DGEES, but is fairly complex and will take me some
time to learn/implement. Furthermore, the implementation of
methods 2 or 3 require the double-shift method 1 anyway, so
it was not a waste of time to code up the double-shift method.

I propose adding the double-shift method to gsl (assuming it
passes any further tests needed) which would be a perfectly
fine eigenvalue solver (EISPACK used it for a number of years)
until I (or someone else) can find the time to implement method 3.

Attached is the latest (final? :)) patch.

Patrick Alken
-------------- next part --------------
diff -urN /home/palken/tmp2/gsl-1.8/eigen/balance.c ./eigen/balance.c
--- /home/palken/tmp2/gsl-1.8/eigen/balance.c	1969-12-31 17:00:00.000000000 -0700
+++ ./eigen/balance.c	2006-05-25 11:24:48.000000000 -0600
@@ -0,0 +1,117 @@
+/* eigen/balance.c
+ * 
+ * Copyright (C) 2006 Patrick Alken
+ * 
+ * This program is free software; you can redistribute it and/or modify
+ * it under the terms of the GNU General Public License as published by
+ * the Free Software Foundation; either version 2 of the License, or (at
+ * your option) any later version.
+ * 
+ * This program is distributed in the hope that it will be useful, but
+ * WITHOUT ANY WARRANTY; without even the implied warranty of
+ * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
+ * General Public License for more details.
+ * 
+ * You should have received a copy of the GNU General Public License
+ * along with this program; if not, write to the Free Software
+ * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
+ */
+
+#include <stdlib.h>
+#include <math.h>
+
+#include "balance.h"
+
+/*
+balance_matrix()
+  Balance a given matrix by applying a diagonal matrix
+similarity transformation so that the rows and columns
+of the new matrix have norms which are the same order of
+magnitude. This is necessary for the unsymmetric eigenvalue
+problem since the calculation can become numerically unstable
+for unbalanced matrices.
+
+See Golub & Van Loan, "Matrix Computations" (3rd ed), Section 7.5.7
+
+and
+
+Numerical Recipes in C, Section 11.5
+*/
+
+void
+balance_matrix(gsl_matrix * A)
+{
+  double row_norm,
+         col_norm;
+  int not_converged;
+  const size_t N = A->size1;
+
+  not_converged = 1;
+
+  while (not_converged)
+    {
+      size_t i, j;
+      double g, f, s;
+
+      not_converged = 0;
+
+      for (i = 0; i < N; ++i)
+        {
+          row_norm = 0.0;
+          col_norm = 0.0;
+
+          for (j = 0; j < N; ++j)
+            {
+              if (j != i)
+                {
+                  col_norm += fabs(gsl_matrix_get(A, j, i));
+                  row_norm += fabs(gsl_matrix_get(A, i, j));
+                }
+            }
+
+          if ((col_norm == 0.0) || (row_norm == 0.0))
+            {
+              continue;
+            }
+
+          g = row_norm / FLOAT_RADIX;
+          f = 1.0;
+          s = col_norm + row_norm;
+
+          /*
+           * find the integer power of the machine radix which
+           * comes closest to balancing the matrix
+           */
+          while (col_norm < g)
+            {
+              f *= FLOAT_RADIX;
+              col_norm *= FLOAT_RADIX_SQ;
+            }
+
+          g = row_norm * FLOAT_RADIX;
+
+          while (col_norm > g)
+            {
+              f /= FLOAT_RADIX;
+              col_norm /= FLOAT_RADIX_SQ;
+            }
+
+          if ((row_norm + col_norm) < 0.95 * s * f)
+            {
+              not_converged = 1;
+
+              g = 1.0 / f;
+
+              /* apply similarity transformation */
+              for (j = 0; j < N; ++j)
+                {
+                  gsl_matrix_set(A, i, j, gsl_matrix_get(A, i, j) * g);
+                }
+              for (j = 0; j < N; ++j)
+                {
+                  gsl_matrix_set(A, j, i, gsl_matrix_get(A, j, i) * f);
+                }
+            }
+        }
+    }
+}
diff -urN /home/palken/tmp2/gsl-1.8/eigen/balance.h ./eigen/balance.h
--- /home/palken/tmp2/gsl-1.8/eigen/balance.h	1969-12-31 17:00:00.000000000 -0700
+++ ./eigen/balance.h	2006-05-25 11:24:48.000000000 -0600
@@ -0,0 +1,32 @@
+/* eigen/balance.h
+ * 
+ * Copyright (C) 2006 Patrick Alken
+ * 
+ * This program is free software; you can redistribute it and/or modify
+ * it under the terms of the GNU General Public License as published by
+ * the Free Software Foundation; either version 2 of the License, or (at
+ * your option) any later version.
+ * 
+ * This program is distributed in the hope that it will be useful, but
+ * WITHOUT ANY WARRANTY; without even the implied warranty of
+ * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
+ * General Public License for more details.
+ * 
+ * You should have received a copy of the GNU General Public License
+ * along with this program; if not, write to the Free Software
+ * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
+ */
+
+#ifndef __GSL_BALANCE_H__
+#define __GSL_BALANCE_H__
+
+#include <gsl/gsl_matrix.h>
+
+/* floating point radix */
+#define FLOAT_RADIX       2.0
+
+#define FLOAT_RADIX_SQ    (FLOAT_RADIX * FLOAT_RADIX)
+
+void balance_matrix(gsl_matrix * A);
+
+#endif /* __GSL_BALANCE_H__ */
diff -urN /home/palken/tmp2/gsl-1.8/eigen/gsl_eigen.h ./eigen/gsl_eigen.h
--- /home/palken/tmp2/gsl-1.8/eigen/gsl_eigen.h	2005-06-26 07:25:34.000000000 -0600
+++ ./eigen/gsl_eigen.h	2006-06-12 15:03:26.000000000 -0600
@@ -58,6 +58,18 @@
 int gsl_eigen_symmv (gsl_matrix * A, gsl_vector * eval, gsl_matrix * evec, gsl_eigen_symmv_workspace * w);
 
 typedef struct {
+  size_t size;    /* size of matrices */
+  unsigned int max_iterations; /* max iterations since last eigenvalue found */
+  gsl_vector *v2; /* temporary 2x1 vector */
+  gsl_vector *v3; /* temporary 3x1 vector */
+} gsl_eigen_unsymm_workspace;
+
+gsl_eigen_unsymm_workspace * gsl_eigen_unsymm_alloc (const size_t n);
+void gsl_eigen_unsymm_free (gsl_eigen_unsymm_workspace * w);
+int gsl_eigen_unsymm (gsl_matrix * A, gsl_vector_complex * eval,
+                      gsl_eigen_unsymm_workspace * w);
+
+typedef struct {
   size_t size;
   double * d;
   double * sd;
diff -urN /home/palken/tmp2/gsl-1.8/eigen/Makefile.am ./eigen/Makefile.am
--- /home/palken/tmp2/gsl-1.8/eigen/Makefile.am	2004-09-11 07:45:45.000000000 -0600
+++ ./eigen/Makefile.am	2006-05-25 11:24:48.000000000 -0600
@@ -3,7 +3,7 @@
 check_PROGRAMS = test
 
 pkginclude_HEADERS = gsl_eigen.h
-libgsleigen_la_SOURCES =  jacobi.c symm.c symmv.c herm.c hermv.c sort.c
+libgsleigen_la_SOURCES =  jacobi.c symm.c symmv.c unsymm.c herm.c hermv.c sort.c balance.c
 INCLUDES= -I$(top_builddir)
 
 noinst_HEADERS =  qrstep.c 
diff -urN /home/palken/tmp2/gsl-1.8/eigen/unsymm.c ./eigen/unsymm.c
--- /home/palken/tmp2/gsl-1.8/eigen/unsymm.c	1969-12-31 17:00:00.000000000 -0700
+++ ./eigen/unsymm.c	2006-06-12 15:05:30.000000000 -0600
@@ -0,0 +1,552 @@
+/* eigen/unsymm.c
+ * 
+ * Copyright (C) 2006 Patrick Alken
+ * 
+ * This program is free software; you can redistribute it and/or modify
+ * it under the terms of the GNU General Public License as published by
+ * the Free Software Foundation; either version 2 of the License, or (at
+ * your option) any later version.
+ * 
+ * This program is distributed in the hope that it will be useful, but
+ * WITHOUT ANY WARRANTY; without even the implied warranty of
+ * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
+ * General Public License for more details.
+ * 
+ * You should have received a copy of the GNU General Public License
+ * along with this program; if not, write to the Free Software
+ * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
+ */
+
+#include <stdlib.h>
+#include <math.h>
+#include <gsl/gsl_eigen.h>
+#include <gsl/gsl_linalg.h>
+#include <gsl/gsl_math.h>
+#include <gsl/gsl_blas.h>
+#include <gsl/gsl_vector.h>
+#include <gsl/gsl_vector_complex.h>
+#include <gsl/gsl_matrix.h>
+
+#include "balance.h"
+
+/*
+ * This module computes the eigenvalues of a real unsymmetric
+ * matrix, using the QR decomposition.
+ *
+ * See Golub & Van Loan, "Matrix Computations" (3rd ed),
+ * algorithm 7.5.2
+ */
+
+inline static size_t schur_decomp(gsl_matrix * H, size_t top,
+                                  size_t bot, gsl_vector_complex * eval,
+                                  size_t evidx,
+                                  size_t * nit,
+                                  gsl_eigen_unsymm_workspace * w);
+inline static size_t zero_subdiag_small_elements(gsl_matrix * A);
+static inline int francis_qrstep(gsl_matrix * H,
+                                 gsl_eigen_unsymm_workspace * w,
+                                 double s, double t);
+static void get_2b2_eigenvalues(gsl_matrix * A, gsl_complex * e1,
+                                gsl_complex * e2);
+
+/*
+gsl_eigen_unsymm_alloc()
+
+Allocate a workspace for solving the unsymmetric
+eigenvalue/eigenvector problem
+
+Inputs: n - size of matrix
+
+Return: pointer to workspace
+*/
+
+gsl_eigen_unsymm_workspace *
+gsl_eigen_unsymm_alloc(const size_t n)
+{
+  gsl_eigen_unsymm_workspace *w;
+
+  if (n == 0)
+    {
+      GSL_ERROR_NULL ("matrix dimension must be positive integer",
+                      GSL_EINVAL);
+    }
+
+  w = ((gsl_eigen_unsymm_workspace *)
+       malloc (sizeof (gsl_eigen_unsymm_workspace)));
+
+  if (w == 0)
+    {
+      GSL_ERROR_NULL ("failed to allocate space for workspace", GSL_ENOMEM);
+    }
+
+  w->size = n;
+  w->max_iterations = 30 * n;
+  w->v2 = gsl_vector_alloc(2);
+  w->v3 = gsl_vector_alloc(3);
+
+  return (w);
+}
+
+void
+gsl_eigen_unsymm_free (gsl_eigen_unsymm_workspace * w)
+{
+  gsl_vector_free(w->v2);
+
+  gsl_vector_free(w->v3);
+
+  free(w);
+}
+
+/*
+gsl_eigen_unsymm()
+
+Solve the unsymmetric eigenvalue problem
+
+A x = \lambda x
+
+for the eigenvalues \lambda using algorithm 7.5.2 of
+Golub & Van Loan, "Matrix Computations" (3rd ed)
+
+Inputs: A    - matrix
+        eval - where to store eigenvalues
+        w    - workspace
+
+Notes: On output, A contains the upper block triangular Schur
+       decomposition.
+*/
+
+int
+gsl_eigen_unsymm (gsl_matrix * A, gsl_vector_complex * eval,
+                  gsl_eigen_unsymm_workspace * w)
+{
+  /* check matrix and vector sizes */
+
+  if (A->size1 != A->size2)
+    {
+      GSL_ERROR ("matrix must be square to compute eigenvalues", GSL_ENOTSQR);
+    }
+  else if (eval->size != A->size1)
+    {
+      GSL_ERROR ("eigenvalue vector must match matrix size", GSL_EBADLEN);
+    }
+  else
+    {
+      size_t N;
+      gsl_complex lambda1, lambda2; /* eigenvalues */
+      size_t nit;
+
+      N = A->size1;
+
+      /* special cases */
+      if (N == 1)
+      {
+        GSL_SET_COMPLEX(&lambda1, gsl_matrix_get(A, 0, 0), 0.0);
+        gsl_vector_complex_set(eval, 0, lambda1);
+        return GSL_SUCCESS;
+      }
+
+      if (N == 2)
+      {
+        /*
+         * The 2x2 case is special since the matrix is already
+         * in upper quasi-triangular form so no Schur decomposition
+         * is necessary
+         */
+        get_2b2_eigenvalues(A, &lambda1, &lambda2);
+        gsl_vector_complex_set(eval, 0, lambda1);
+        gsl_vector_complex_set(eval, 1, lambda2);
+        return GSL_SUCCESS;
+      }
+
+      /* balance the matrix */
+      balance_matrix(A);
+
+      /* compute the Hessenberg reduction of A */
+      gsl_linalg_hessenberg(A);
+
+      nit = 0;
+
+      /*
+       * compute Schur decomposition of A and store eigenvalues
+       * into eval
+       */
+      schur_decomp(A, 0, N - 1, eval, 0, &nit, w);
+
+      if (nit > w->max_iterations)
+        {
+          GSL_ERROR("maximum iterations exceeded", GSL_EMAXITER);
+        }
+
+      return GSL_SUCCESS;
+    }
+} /* gsl_eigen_unsymm() */
+
+/********************************************
+ *           INTERNAL ROUTINES              *
+ ********************************************/
+
+/*
+schur_decomp()
+  Compute the Schur decomposition of the submatrix of H
+starting from (top, top) to (bot, bot)
+
+Inputs: H     - hessenberg matrix
+        top   - top index
+        bot   - bottom index
+        eval  - where to store eigenvalues
+        evidx - index into eval
+        nit   - running total of number of QR iterations since
+                we last found an eigenvalue
+                (must be initialized before calling this function)
+        w     - workspace
+
+Return: number of eigenvalues found
+*/
+
+inline static size_t
+schur_decomp(gsl_matrix * H, size_t top, size_t bot,
+             gsl_vector_complex * eval, size_t evidx,
+             size_t * nit,
+             gsl_eigen_unsymm_workspace * w)
+{
+  gsl_matrix_view m;
+  size_t N;    /* size of matrix */
+  double s, t; /* shifts */
+  size_t nev;  /* number of eigenvalues found so far */
+  size_t q;
+  gsl_complex lambda1, /* eigenvalues */
+              lambda2;
+
+  N = bot - top + 1;
+
+  if (N == 1)
+    {
+      GSL_SET_COMPLEX(&lambda1, gsl_matrix_get(H, 0, 0), 0.0);
+      gsl_vector_complex_set(eval, evidx, lambda1);
+      *nit = 0;
+      return 1;
+    }
+  else if (N == 2)
+    {
+      get_2b2_eigenvalues(H, &lambda1, &lambda2);
+      gsl_vector_complex_set(eval, evidx, lambda1);
+      gsl_vector_complex_set(eval, evidx + 1, lambda2);
+      *nit = 0;
+      return 2;
+    }
+
+  m = gsl_matrix_submatrix(H, top, top, N, N);
+
+  nev = 0;
+  while ((N > 2) && ((*nit)++ < w->max_iterations))
+    {
+      if ((*nit == 10) || (*nit == 20))
+        {
+          /*
+           * We have gone 10 or 20 iterations without finding
+           * a new eigenvalue, try a new choice of shifts.
+           * See Numerical Recipes in C, eq 11.6.27
+           */
+          t = fabs(gsl_matrix_get(&m.matrix, N - 1, N - 2)) +
+              fabs(gsl_matrix_get(&m.matrix, N - 2, N - 3));
+          s = 1.5 * t;
+          t *= t;
+        }
+      else
+        {
+          /* s = a_1 + a_2 = m_{mm} + m_{nn} where m = n - 1 */
+          s = gsl_matrix_get(&m.matrix, N - 2, N - 2) +
+              gsl_matrix_get(&m.matrix, N - 1, N - 1);
+
+          /* t = a_1 * a_2 = m_{mm} * m_{nn} - m_{mn} * m_{nm} */
+          t = (gsl_matrix_get(&m.matrix, N - 2, N - 2) *
+               gsl_matrix_get(&m.matrix, N - 1, N - 1)) -
+              (gsl_matrix_get(&m.matrix, N - 2, N - 1) *
+               gsl_matrix_get(&m.matrix, N - 1, N - 2));
+        }
+
+      francis_qrstep(&m.matrix, w, s, t);
+      q = zero_subdiag_small_elements(&m.matrix);
+
+      if (q == 0)
+        {
+          /* no small subdiagonal element found */
+          continue;
+        }
+
+      if (q == (N - 1))
+        {
+          /*
+           * the last subdiagonal element of the matrix is 0 -
+           * m_{NN} is a real eigenvalue
+           */
+          GSL_SET_COMPLEX(&lambda1,
+                          gsl_matrix_get(&m.matrix, q, q), 0.0);
+          gsl_vector_complex_set(eval, evidx + nev++, lambda1);
+          *nit = 0;
+
+          --N;
+          m = gsl_matrix_submatrix(&m.matrix, 0, 0, N, N);
+        }
+      else if (q == (N - 2))
+        {
+          gsl_matrix_view v;
+
+          /*
+           * The bottom right 2x2 block of m is an eigenvalue
+           * system
+           */
+
+          v = gsl_matrix_submatrix(&m.matrix, q, q, 2, 2);
+          get_2b2_eigenvalues(&v.matrix, &lambda1, &lambda2);
+          gsl_vector_complex_set(eval, evidx + nev++, lambda1);
+          gsl_vector_complex_set(eval, evidx + nev++, lambda2);
+          *nit = 0;
+
+          N -= 2;
+          m = gsl_matrix_submatrix(&m.matrix, 0, 0, N, N);
+        }
+      else if (q == 1)
+        {
+          /* the first matrix element is an eigenvalue */
+          GSL_SET_COMPLEX(&lambda1,
+                          gsl_matrix_get(&m.matrix, 0, 0), 0.0);
+          gsl_vector_complex_set(eval, evidx + nev++, lambda1);
+          *nit = 0;
+
+          --N;
+          m = gsl_matrix_submatrix(&m.matrix, 1, 1, N, N);
+        }
+      else if (q == 2)
+        {
+          gsl_matrix_view v;
+
+          /* the upper left 2x2 block is an eigenvalue system */
+
+          v = gsl_matrix_submatrix(&m.matrix, 0, 0, 2, 2);
+          get_2b2_eigenvalues(&v.matrix, &lambda1, &lambda2);
+          gsl_vector_complex_set(eval, evidx + nev++, lambda1);
+          gsl_vector_complex_set(eval, evidx + nev++, lambda2);
+          *nit = 0;
+
+          N -= 2;
+          m = gsl_matrix_submatrix(&m.matrix, 2, 2, N, N);
+        }
+      else
+        {
+          /*
+           * There is a zero element on the subdiagonal somewhere
+           * in the middle of the matrix - we can now operate
+           * separately on the two submatrices split by this
+           * element. q is the row index of the zero element.
+           */
+
+          /* operate on lower right (N - q)x(N - q) block first */
+          nev += schur_decomp(&m.matrix,
+                              q,
+                              N - 1,
+                              eval,
+                              evidx + nev,
+                              nit,
+                              w);
+
+          /* operate on upper left qxq block */
+          nev += schur_decomp(&m.matrix,
+                              0,
+                              q - 1,
+                              eval,
+                              evidx + nev,
+                              nit,
+                              w);
+          N = 0;
+        }
+    }
+
+  if (N == 1)
+    {
+      GSL_SET_COMPLEX(&lambda1, gsl_matrix_get(&m.matrix, 0, 0), 0.0);
+      gsl_vector_complex_set(eval, evidx + nev++, lambda1);
+      *nit = 0;
+    }
+  else if (N == 2)
+    {
+      get_2b2_eigenvalues(&m.matrix, &lambda1, &lambda2);
+      gsl_vector_complex_set(eval, evidx + nev++, lambda1);
+      gsl_vector_complex_set(eval, evidx + nev++, lambda2);
+      *nit = 0;
+    }
+
+  return (nev);
+}
+
+/*
+zero_subdiag_small_elements()
+  Sets to zero all elements on the subdiaganal of a matrix A
+which satisfy
+
+|A_{i,i-1}| <= eps * (|A_{i,i}| + |A_{i-1,i-1}|)
+
+Inputs: A - matrix (must be at least 3x3)
+
+Return: row index of small subdiagonal element or 0 if not found
+*/
+
+inline static size_t
+zero_subdiag_small_elements(gsl_matrix * A)
+{
+  const size_t N = A->size1;
+  size_t i;
+  double dpel = gsl_matrix_get(A, N - 2, N - 2);
+
+  for (i = N - 1; i > 0; --i)
+    {
+      double sel = gsl_matrix_get(A, i, i - 1);
+      double del = gsl_matrix_get(A, i, i);
+
+      if ((sel == 0.0) ||
+          (fabs(sel) < GSL_DBL_EPSILON * (fabs(del) + fabs(dpel))))
+        {
+          gsl_matrix_set(A, i, i - 1, 0.0);
+          return (i);
+        }
+
+      dpel = del;
+    }
+
+  return (0);
+}
+
+/*
+francis_qrstep()
+  Perform a Francis QR step.
+
+See Golub & Van Loan, "Matrix Computations" (3rd ed),
+algorithm 7.5.1
+
+Inputs: H - unreduced upper Hessenberg matrix
+        w - workspace
+        s - sum of current shifts (a_1 + a_2)
+        t - product of shifts (a_1 * a_2)
+*/
+
+static inline int
+francis_qrstep(gsl_matrix * H, gsl_eigen_unsymm_workspace * w,
+               double s, double t)
+{
+  const size_t N = H->size1;
+  double x, y, z;
+  double scale;
+  size_t i;
+  gsl_matrix_view m;
+  double tau_i;
+  size_t q, r;
+
+  x = gsl_matrix_get(H, 0, 0) * gsl_matrix_get(H, 0, 0) +
+      gsl_matrix_get(H, 0, 1) * gsl_matrix_get(H, 1, 0) -
+      s*gsl_matrix_get(H, 0, 0) + t;
+  y = gsl_matrix_get(H, 1, 0) *
+      (gsl_matrix_get(H, 0, 0) + gsl_matrix_get(H, 1, 1) - s);
+  z = gsl_matrix_get(H, 1, 0) * gsl_matrix_get(H, 2, 1);
+
+  scale = fabs(x) + fabs(y) + fabs(z);
+  if (scale != 0.0)
+    {
+      /* scale to prevent overflow or underflow */
+      x /= scale;
+      y /= scale;
+      z /= scale;
+    }
+
+  for (i = 0; i < N - 2; ++i)
+    {
+      gsl_vector_set(w->v3, 0, x);
+      gsl_vector_set(w->v3, 1, y);
+      gsl_vector_set(w->v3, 2, z);
+      tau_i = gsl_linalg_householder_transform(w->v3);
+
+      if (tau_i != 0.0)
+        {
+          /* q = max(1, i - 1) */
+          q = (1 > ((int)i - 1)) ? 0 : (i - 1);
+
+          /* apply left householder matrix (I - tau_i v v') to H */
+          m = gsl_matrix_submatrix(H, i, q, 3, N - q);
+          gsl_linalg_householder_hm(tau_i, w->v3, &m.matrix);
+
+          /* r = min(i + 3, N - 1) */
+          r = ((i + 3) < (N - 1)) ? (i + 3) : (N - 1);
+
+          /* apply right householder matrix (I - tau_i v v') to H */
+          m = gsl_matrix_submatrix(H, 0, i, r + 1, 3);
+          gsl_linalg_householder_mh(tau_i, w->v3, &m.matrix);
+        }
+
+      x = gsl_matrix_get(H, i + 1, i);
+      y = gsl_matrix_get(H, i + 2, i);
+      if (i < (N - 3))
+        {
+          z = gsl_matrix_get(H, i + 3, i);
+        }
+
+      scale = fabs(x) + fabs(y) + fabs(z);
+      if (scale != 0.0)
+        {
+          /* scale to prevent overflow or underflow */
+          x /= scale;
+          y /= scale;
+          z /= scale;
+        }
+    }
+
+  gsl_vector_set(w->v2, 0, x);
+  gsl_vector_set(w->v2, 1, y);
+  tau_i = gsl_linalg_householder_transform(w->v2);
+
+  m = gsl_matrix_submatrix(H, N - 2, N - 3, 2, 3);
+  gsl_linalg_householder_hm(tau_i, w->v2, &m.matrix);
+
+  m = gsl_matrix_submatrix(H, 0, N - 2, N, 2);
+  gsl_linalg_householder_mh(tau_i, w->v2, &m.matrix);
+
+  return GSL_SUCCESS;
+}
+
+/*
+get_2b2_eigenvalues()
+  Compute the eigenvalues of a 2x2 real matrix
+
+Inputs: A  - 2x2 matrix
+        e1 - where to store eigenvalue 1
+        e2 - where to store eigenvalue 2
+*/
+
+static void
+get_2b2_eigenvalues(gsl_matrix * A, gsl_complex * e1,
+                    gsl_complex * e2)
+{
+  double discr;      /* discriminant of characteristic poly */
+  double a, b, c, d; /* matrix values */
+
+  a = gsl_matrix_get(A, 0, 0);
+  b = gsl_matrix_get(A, 0, 1);
+  c = gsl_matrix_get(A, 1, 0);
+  d = gsl_matrix_get(A, 1, 1);
+
+  discr = (a + d)*(a + d) - 4.0*(a*d - b*c);
+  if (discr < 0.0)
+    {
+      GSL_SET_REAL(e1, 0.5*(a + d));
+      GSL_SET_REAL(e2, 0.5*(a + d));
+
+      GSL_SET_IMAG(e1, 0.5*sqrt(-discr));
+      GSL_SET_IMAG(e2, -0.5*sqrt(-discr));
+    }
+  else
+    {
+      GSL_SET_REAL(e1, 0.5*(a + d + sqrt(discr)));
+      GSL_SET_REAL(e2, 0.5*(a + d - sqrt(discr)));
+
+      GSL_SET_IMAG(e1, 0.0);
+      GSL_SET_IMAG(e2, 0.0);
+    }
+}
diff -urN /home/palken/tmp2/gsl-1.8/linalg/gsl_linalg.h ./linalg/gsl_linalg.h
--- /home/palken/tmp2/gsl-1.8/linalg/gsl_linalg.h	2005-11-09 14:13:14.000000000 -0700
+++ ./linalg/gsl_linalg.h	2006-05-25 11:24:48.000000000 -0600
@@ -113,6 +113,10 @@
                                        const gsl_vector_complex * v, 
                                        gsl_vector_complex * w);
 
+/* Hessenberg reduction */
+
+int gsl_linalg_hessenberg (gsl_matrix * A);
+
 /* Singular Value Decomposition
 
  * exceptions: 
diff -urN /home/palken/tmp2/gsl-1.8/linalg/hessenberg.c ./linalg/hessenberg.c
--- /home/palken/tmp2/gsl-1.8/linalg/hessenberg.c	1969-12-31 17:00:00.000000000 -0700
+++ ./linalg/hessenberg.c	2006-05-25 11:24:48.000000000 -0600
@@ -0,0 +1,76 @@
+/* linalg/hessenberg.c
+ * 
+ * Copyright (C) 2006 Patrick Alken
+ * 
+ * This program is free software; you can redistribute it and/or modify
+ * it under the terms of the GNU General Public License as published by
+ * the Free Software Foundation; either version 2 of the License, or (at
+ * your option) any later version.
+ * 
+ * This program is distributed in the hope that it will be useful, but
+ * WITHOUT ANY WARRANTY; without even the implied warranty of
+ * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
+ * General Public License for more details.
+ * 
+ * You should have received a copy of the GNU General Public License
+ * along with this program; if not, write to the Free Software
+ * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
+ */
+
+#include <gsl/gsl_linalg.h>
+#include <gsl/gsl_matrix.h>
+#include <gsl/gsl_vector.h>
+
+/*
+gsl_linalg_hessenberg()
+  Compute the Householder reduction to Hessenberg form of a
+square matrix A.
+
+See Golub & Van Loan, "Matrix Computations" (3rd ed), algorithm
+7.4.2
+*/
+
+int
+gsl_linalg_hessenberg(gsl_matrix * A)
+{
+  if (A->size1 != A->size2)
+    {
+      GSL_ERROR ("Hessenberg reduction requires square matrix",
+                 GSL_ENOTSQR);
+    }
+  else
+    {
+      const size_t N = A->size1;
+      size_t i, j;
+      gsl_vector_view v;
+      gsl_vector *v_copy;
+      gsl_matrix_view m;
+      double tau_i; /* beta in algorithm 7.4.2 */
+
+      v_copy = gsl_vector_alloc(N);
+
+      for (i = 0; i < N - 2; ++i)
+        {
+          /* make a copy of A(i + 1:n, i) */
+
+          v = gsl_matrix_column(A, i);
+          gsl_vector_memcpy(v_copy, &v.vector);
+          v = gsl_vector_subvector(v_copy, i + 1, N - (i + 1));
+
+          /* compute householder transformation of A(i+1:n,i) */
+          tau_i = gsl_linalg_householder_transform(&v.vector);
+
+          /* apply left householder matrix (I - tau_i v v') to A */
+          m = gsl_matrix_submatrix(A, i + 1, i, N - (i + 1), N - i);
+          gsl_linalg_householder_hm(tau_i, &v.vector, &m.matrix);
+
+          /* apply right householder matrix (I - tau_i v v') to A */
+          m = gsl_matrix_submatrix(A, 0, i + 1, N, N - (i + 1));
+          gsl_linalg_householder_mh(tau_i, &v.vector, &m.matrix);
+        }
+
+      gsl_vector_free(v_copy);
+
+      return GSL_SUCCESS;
+    }
+}
diff -urN /home/palken/tmp2/gsl-1.8/linalg/Makefile.am ./linalg/Makefile.am
--- /home/palken/tmp2/gsl-1.8/linalg/Makefile.am	2004-09-13 12:17:04.000000000 -0600
+++ ./linalg/Makefile.am	2006-05-25 11:24:48.000000000 -0600
@@ -4,7 +4,7 @@
 
 INCLUDES= -I$(top_builddir)
 
-libgsllinalg_la_SOURCES = multiply.c exponential.c tridiag.c tridiag.h lu.c luc.c hh.c qr.c qrpt.c lq.c ptlq.c svd.c householder.c householdercomplex.c cholesky.c symmtd.c hermtd.c bidiag.c balance.c
+libgsllinalg_la_SOURCES = multiply.c exponential.c tridiag.c tridiag.h lu.c luc.c hh.c qr.c qrpt.c lq.c ptlq.c svd.c householder.c householdercomplex.c hessenberg.c cholesky.c symmtd.c hermtd.c bidiag.c balance.c
 
 noinst_HEADERS =  givens.c apply_givens.c svdstep.c tridiag.h 
 


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