Now using 2-norm to compute vector norms of row and column for balancing algorithm.

This seems to fix problems with balancing causing very large backward error for certain
Hessenberg matrices, including the often cited example of Watkins.

diff --git a/SRC/cgebal.f b/SRC/cgebal.f
index 4bb5a2f0be22096191e3ccfc87a70d49ce8841b8..47207ea3084d55eda704a9d065e7ae7c73ee6616 100644 (file)
--- a/SRC/cgebal.f
+++ b/SRC/cgebal.f
@@ -195,8 +195,8 @@
 *     .. External Functions ..
       LOGICAL            SISNAN, LSAME
       INTEGER            ICAMAX
-      REAL               SLAMCH
-      EXTERNAL           SISNAN, LSAME, ICAMAX, SLAMCH
+      REAL               SLAMCH, SCNRM2
+      EXTERNAL           SISNAN, LSAME, ICAMAX, SLAMCH, SCNRM2
 *     ..
 *     .. External Subroutines ..
       EXTERNAL           CSSCAL, CSWAP, XERBLA
@@ -325,15 +325,9 @@
       NOCONV = .FALSE.
 *
       DO 200 I = K, L
-         C = ZERO
-         R = ZERO
-*
-         DO 150 J = K, L
-            IF( J.EQ.I )
-     $         GO TO 150
-            C = C + CABS1( A( J, I ) )
-            R = R + CABS1( A( I, J ) )
-  150    CONTINUE
+*
+         C = SCNRM2( L-K+1, A( K, I ), 1 )
+         R = SCNRM2( L-K+1, A( I , K ), LDA )
          ICA = ICAMAX( L, A( 1, I ), 1 )
          CA = ABS( A( ICA, I ) )
          IRA = ICAMAX( N-K+1, A( I, K ), LDA )

diff --git a/SRC/dgebal.f b/SRC/dgebal.f
index 5d7ed035c3465717eecfbd6fc99af40ea9cb47e9..fe4c156bdd182db7d1c9a0229a4ece70cbc20247 100644 (file)
--- a/SRC/dgebal.f
+++ b/SRC/dgebal.f
@@ -192,8 +192,8 @@
 *     .. External Functions ..
       LOGICAL            DISNAN, LSAME
       INTEGER            IDAMAX
-      DOUBLE PRECISION   DLAMCH
-      EXTERNAL           DISNAN, LSAME, IDAMAX, DLAMCH
+      DOUBLE PRECISION   DLAMCH, DNRM2
+      EXTERNAL           DISNAN, LSAME, IDAMAX, DLAMCH, DNRM2
 *     ..
 *     .. External Subroutines ..
       EXTERNAL           DSCAL, DSWAP, XERBLA
@@ -312,19 +312,14 @@
       SFMAX1 = ONE / SFMIN1
       SFMIN2 = SFMIN1*SCLFAC
       SFMAX2 = ONE / SFMIN2
+*
   140 CONTINUE
       NOCONV = .FALSE.
 *
       DO 200 I = K, L
-         C = ZERO
-         R = ZERO
-*
-         DO 150 J = K, L
-            IF( J.EQ.I )
-     $         GO TO 150
-            C = C + ABS( A( J, I ) )
-            R = R + ABS( A( I, J ) )
-  150    CONTINUE
+*
+         C = DNRM2( L-K+1, A( K, I ), 1 )
+         R = DNRM2( L-K+1, A( I, K ), LDA )
          ICA = IDAMAX( L, A( 1, I ), 1 )
          CA = ABS( A( ICA, I ) )
          IRA = IDAMAX( N-K+1, A( I, K ), LDA )

diff --git a/SRC/sgebal.f b/SRC/sgebal.f
index 853ff20e27ddacb5603a7a433bf21f3b10aa1d4a..de948488353d023023abc953af394f0d5c9d257a 100644 (file)
--- a/SRC/sgebal.f
+++ b/SRC/sgebal.f
@@ -192,8 +192,8 @@
 *     .. External Functions ..
       LOGICAL            SISNAN, LSAME
       INTEGER            ISAMAX
-      REAL               SLAMCH
-      EXTERNAL           SISNAN, LSAME, ISAMAX, SLAMCH
+      REAL               SLAMCH, SNRM2
+      EXTERNAL           SISNAN, LSAME, ISAMAX, SLAMCH, SNRM2
 *     ..
 *     .. External Subroutines ..
       EXTERNAL           SSCAL, SSWAP, XERBLA
@@ -316,15 +316,9 @@
       NOCONV = .FALSE.
 *
       DO 200 I = K, L
-         C = ZERO
-         R = ZERO
-*
-         DO 150 J = K, L
-            IF( J.EQ.I )
-     $         GO TO 150
-            C = C + ABS( A( J, I ) )
-            R = R + ABS( A( I, J ) )
-  150    CONTINUE
+*
+         C = SNRM2( L-K+1, A( K, I ), 1 )
+         R = SNRM2( L-K+1, A( I, K ), LDA )
          ICA = ISAMAX( L, A( 1, I ), 1 )
          CA = ABS( A( ICA, I ) )
          IRA = ISAMAX( N-K+1, A( I, K ), LDA )

diff --git a/SRC/zgebal.f b/SRC/zgebal.f
index 9c90f0b4b5f4b2d4f4c26315ab2cafb0d60f6bab..aa4f6d0e7c9fc42bc508589bd25f48af9d76fc9d 100644 (file)
--- a/SRC/zgebal.f
+++ b/SRC/zgebal.f
@@ -194,8 +194,8 @@
 *     .. External Functions ..
       LOGICAL            DISNAN, LSAME
       INTEGER            IZAMAX
-      DOUBLE PRECISION   DLAMCH
-      EXTERNAL           DISNAN, LSAME, IZAMAX, DLAMCH
+      DOUBLE PRECISION   DLAMCH, DZNRM2
+      EXTERNAL           DISNAN, LSAME, IZAMAX, DLAMCH, DZNRM2
 *     ..
 *     .. External Subroutines ..
       EXTERNAL           XERBLA, ZDSCAL, ZSWAP
@@ -324,15 +324,9 @@
       NOCONV = .FALSE.
 *
       DO 200 I = K, L
-         C = ZERO
-         R = ZERO
-*
-         DO 150 J = K, L
-            IF( J.EQ.I )
-     $         GO TO 150
-            C = C + CABS1( A( J, I ) )
-            R = R + CABS1( A( I, J ) )
-  150    CONTINUE
+*
+         C = DZNRM2( L-K+1, A( K, I ), 1 )
+         R = DZNRM2( L-K+1, A( I, K ), LDA )
          ICA = IZAMAX( L, A( 1, I ), 1 )
          CA = ABS( A( ICA, I ) )
          IRA = IZAMAX( N-K+1, A( I, K ), LDA )