Update Frobenius norms for better accuracy.

SRC/CMakeLists.txt

@@ -150,7 +150,8 @@ set(SLASRC
    stplqt.f stplqt2.f stpmlqt.f
    ssytrd_2stage.f ssytrd_sy2sb.f ssytrd_sb2st.F ssb2st_kernels.f
    ssyevd_2stage.f ssyev_2stage.f ssyevx_2stage.f ssyevr_2stage.f
-   ssbev_2stage.f ssbevx_2stage.f ssbevd_2stage.f ssygv_2stage.f)
+   ssbev_2stage.f ssbevx_2stage.f ssbevd_2stage.f ssygv_2stage.f
+   scombssq.f)
 
 set(DSLASRC spotrs.f sgetrs.f spotrf.f sgetrf.f)
 
@@ -341,7 +342,8 @@ set(DLASRC
    dtplqt.f dtplqt2.f dtpmlqt.f
    dsytrd_2stage.f dsytrd_sy2sb.f dsytrd_sb2st.F dsb2st_kernels.f
    dsyevd_2stage.f dsyev_2stage.f dsyevx_2stage.f dsyevr_2stage.f
-   dsbev_2stage.f dsbevx_2stage.f dsbevd_2stage.f dsygv_2stage.f)
+   dsbev_2stage.f dsbevx_2stage.f dsbevd_2stage.f dsygv_2stage.f
+   dcombssq.f)
 
 set(DXLASRC dgesvxx.f dgerfsx.f dla_gerfsx_extended.f dla_geamv.f
    dla_gercond.f dla_gerpvgrw.f dsysvxx.f dsyrfsx.f

SRC/Makefile

@@ -174,7 +174,7 @@ SLASRC = \
    ssytrd_2stage.o ssytrd_sy2sb.o ssytrd_sb2st.o ssb2st_kernels.o \
    ssyevd_2stage.o ssyev_2stage.o ssyevx_2stage.o ssyevr_2stage.o \
    ssbev_2stage.o ssbevx_2stage.o ssbevd_2stage.o ssygv_2stage.o \
-   sgesvdq.o
+   sgesvdq.o scombssq.o
 
 DSLASRC = spotrs.o sgetrs.o spotrf.o sgetrf.o
 
@@ -373,7 +373,7 @@ DLASRC = \
    dsytrd_2stage.o dsytrd_sy2sb.o dsytrd_sb2st.o dsb2st_kernels.o \
    dsyevd_2stage.o dsyev_2stage.o dsyevx_2stage.o dsyevr_2stage.o \
    dsbev_2stage.o dsbevx_2stage.o dsbevd_2stage.o dsygv_2stage.o \
-   dgesvdq.o
+   dgesvdq.o dcombssq.o
 
 ifdef USEXBLAS
 DXLASRC = dgesvxx.o dgerfsx.o dla_gerfsx_extended.o dla_geamv.o \

SRC/clangb.f

@@ -130,6 +130,7 @@ REAL             FUNCTION CLANGB( NORM, N, KL, KU, AB, LDAB,
 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
 *     December 2016
 *
+      IMPLICIT NONE
 *     .. Scalar Arguments ..
       CHARACTER          NORM
       INTEGER            KL, KU, LDAB, N
@@ -147,14 +148,17 @@ REAL             FUNCTION CLANGB( NORM, N, KL, KU, AB, LDAB,
 *     ..
 *     .. Local Scalars ..
       INTEGER            I, J, K, L
-      REAL               SCALE, SUM, VALUE, TEMP
+      REAL               SUM, VALUE, TEMP
+*     ..
+*     .. Local Arrays ..
+      REAL               SSQ( 2 ), COLSSQ( 2 )
 *     ..
 *     .. External Functions ..
       LOGICAL            LSAME, SISNAN
       EXTERNAL           LSAME, SISNAN
 *     ..
 *     .. External Subroutines ..
-      EXTERNAL           CLASSQ
+      EXTERNAL           CLASSQ, SCOMBSSQ
 *     ..
 *     .. Intrinsic Functions ..
       INTRINSIC          ABS, MAX, MIN, SQRT
@@ -207,15 +211,22 @@ REAL             FUNCTION CLANGB( NORM, N, KL, KU, AB, LDAB,
       ELSE IF( ( LSAME( NORM, 'F' ) ) .OR. ( LSAME( NORM, 'E' ) ) ) THEN
 *
 *        Find normF(A).
+*        SSQ(1) is scale
+*        SSQ(2) is sum-of-squares
+*        For better accuracy, sum each column separately.
 *
-         SCALE = ZERO
-         SUM = ONE
+         SSQ( 1 ) = ZERO
+         SSQ( 2 ) = ONE
          DO 90 J = 1, N
             L = MAX( 1, J-KU )
             K = KU + 1 - J + L
-            CALL CLASSQ( MIN( N, J+KL )-L+1, AB( K, J ), 1, SCALE, SUM )
+            COLSSQ( 1 ) = ZERO
+            COLSSQ( 2 ) = ONE
+            CALL CLASSQ( MIN( N, J+KL )-L+1, AB( K, J ), 1,
+     $                   COLSSQ( 1 ), COLSSQ( 2 ) )
+            CALL SCOMBSSQ( SSQ, COLSSQ )
    90    CONTINUE
-         VALUE = SCALE*SQRT( SUM )
+         VALUE = SSQ( 1 )*SQRT( SSQ( 2 ) )
       END IF
 *
       CLANGB = VALUE

SRC/clange.f

@@ -120,6 +120,7 @@ REAL             FUNCTION CLANGE( NORM, M, N, A, LDA, WORK )
 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
 *     December 2016
 *
+      IMPLICIT NONE
 *     .. Scalar Arguments ..
       CHARACTER          NORM
       INTEGER            LDA, M, N
@@ -137,14 +138,17 @@ REAL             FUNCTION CLANGE( NORM, M, N, A, LDA, WORK )
 *     ..
 *     .. Local Scalars ..
       INTEGER            I, J
-      REAL               SCALE, SUM, VALUE, TEMP
+      REAL               SUM, VALUE, TEMP
+*     ..
+*     .. Local Arrays ..
+      REAL               SSQ( 2 ), COLSSQ( 2 )
 *     ..
 *     .. External Functions ..
       LOGICAL            LSAME, SISNAN
       EXTERNAL           LSAME, SISNAN
 *     ..
 *     .. External Subroutines ..
-      EXTERNAL           CLASSQ
+      EXTERNAL           CLASSQ, SCOMBSSQ
 *     ..
 *     .. Intrinsic Functions ..
       INTRINSIC          ABS, MIN, SQRT
@@ -196,13 +200,19 @@ REAL             FUNCTION CLANGE( NORM, M, N, A, LDA, WORK )
       ELSE IF( ( LSAME( NORM, 'F' ) ) .OR. ( LSAME( NORM, 'E' ) ) ) THEN
 *
 *        Find normF(A).
+*        SSQ(1) is scale
+*        SSQ(2) is sum-of-squares
+*        For better accuracy, sum each column separately.
 *
-         SCALE = ZERO
-         SUM = ONE
+         SSQ( 1 ) = ZERO
+         SSQ( 2 ) = ONE
          DO 90 J = 1, N
-            CALL CLASSQ( M, A( 1, J ), 1, SCALE, SUM )
+            COLSSQ( 1 ) = ZERO
+            COLSSQ( 2 ) = ONE
+            CALL CLASSQ( M, A( 1, J ), 1, COLSSQ( 1 ), COLSSQ( 2 ) )
+            CALL SCOMBSSQ( SSQ, COLSSQ )
    90    CONTINUE
-         VALUE = SCALE*SQRT( SUM )
+         VALUE = SSQ( 1 )*SQRT( SSQ( 2 ) )
       END IF
 *
       CLANGE = VALUE

SRC/clanhb.f

@@ -137,6 +137,7 @@ REAL             FUNCTION CLANHB( NORM, UPLO, N, K, AB, LDAB,
 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
 *     December 2016
 *
+      IMPLICIT NONE
 *     .. Scalar Arguments ..
       CHARACTER          NORM, UPLO
       INTEGER            K, LDAB, N
@@ -154,14 +155,17 @@ REAL             FUNCTION CLANHB( NORM, UPLO, N, K, AB, LDAB,
 *     ..
 *     .. Local Scalars ..
       INTEGER            I, J, L
-      REAL               ABSA, SCALE, SUM, VALUE
+      REAL               ABSA, SUM, VALUE
+*     ..
+*     .. Local Arrays ..
+      REAL               SSQ( 2 ), COLSSQ( 2 )
 *     ..
 *     .. External Functions ..
       LOGICAL            LSAME, SISNAN
       EXTERNAL           LSAME, SISNAN
 *     ..
 *     .. External Subroutines ..
-      EXTERNAL           CLASSQ
+      EXTERNAL           CLASSQ, SCOMBSSQ
 *     ..
 *     .. Intrinsic Functions ..
       INTRINSIC          ABS, MAX, MIN, REAL, SQRT
@@ -233,39 +237,57 @@ REAL             FUNCTION CLANHB( NORM, UPLO, N, K, AB, LDAB,
       ELSE IF( ( LSAME( NORM, 'F' ) ) .OR. ( LSAME( NORM, 'E' ) ) ) THEN
 *
 *        Find normF(A).
+*        SSQ(1) is scale
+*        SSQ(2) is sum-of-squares
+*        For better accuracy, sum each column separately.
+*
+         SSQ( 1 ) = ZERO
+         SSQ( 2 ) = ONE
+*
+*        Sum off-diagonals
 *
-         SCALE = ZERO
-         SUM = ONE
          IF( K.GT.0 ) THEN
             IF( LSAME( UPLO, 'U' ) ) THEN
                DO 110 J = 2, N
+                  COLSSQ( 1 ) = ZERO
+                  COLSSQ( 2 ) = ONE
                   CALL CLASSQ( MIN( J-1, K ), AB( MAX( K+2-J, 1 ), J ),
-     $                         1, SCALE, SUM )
+     $                         1, COLSSQ( 1 ), COLSSQ( 2 ) )
+                  CALL SCOMBSSQ( SSQ, COLSSQ )
   110          CONTINUE
                L = K + 1
             ELSE
                DO 120 J = 1, N - 1
-                  CALL CLASSQ( MIN( N-J, K ), AB( 2, J ), 1, SCALE,
-     $                         SUM )
+                  COLSSQ( 1 ) = ZERO
+                  COLSSQ( 2 ) = ONE
+                  CALL CLASSQ( MIN( N-J, K ), AB( 2, J ), 1,
+     $                         COLSSQ( 1 ), COLSSQ( 2 ) )
+                  CALL SCOMBSSQ( SSQ, COLSSQ )
   120          CONTINUE
                L = 1
             END IF
-            SUM = 2*SUM
+            SSQ( 2 ) = 2*SSQ( 2 )
          ELSE
             L = 1
          END IF
+*
+*        Sum diagonal
+*
+         COLSSQ( 1 ) = ZERO
+         COLSSQ( 2 ) = ONE
          DO 130 J = 1, N
             IF( REAL( AB( L, J ) ).NE.ZERO ) THEN
                ABSA = ABS( REAL( AB( L, J ) ) )
-               IF( SCALE.LT.ABSA ) THEN
-                  SUM = ONE + SUM*( SCALE / ABSA )**2
-                  SCALE = ABSA
+               IF( COLSSQ( 1 ).LT.ABSA ) THEN
+                  COLSSQ( 2 ) = ONE + COLSSQ(2)*( COLSSQ(1) / ABSA )**2
+                  COLSSQ( 1 ) = ABSA
                ELSE
-                  SUM = SUM + ( ABSA / SCALE )**2
+                  COLSSQ( 2 ) = COLSSQ( 2 ) + ( ABSA / COLSSQ( 1 ) )**2
                END IF
             END IF
   130    CONTINUE
-         VALUE = SCALE*SQRT( SUM )
+         CALL SCOMBSSQ( SSQ, COLSSQ )
+         VALUE = SSQ( 1 )*SQRT( SSQ( 2 ) )
       END IF
 *
       CLANHB = VALUE

SRC/clanhe.f

@@ -129,6 +129,7 @@ REAL             FUNCTION CLANHE( NORM, UPLO, N, A, LDA, WORK )
 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
 *     December 2016
 *
+      IMPLICIT NONE
 *     .. Scalar Arguments ..
       CHARACTER          NORM, UPLO
       INTEGER            LDA, N
@@ -146,14 +147,17 @@ REAL             FUNCTION CLANHE( NORM, UPLO, N, A, LDA, WORK )
 *     ..
 *     .. Local Scalars ..
       INTEGER            I, J
-      REAL               ABSA, SCALE, SUM, VALUE
+      REAL               ABSA, SUM, VALUE
+*     ..
+*     .. Local Arrays ..
+      REAL               SSQ( 2 ), COLSSQ( 2 )
 *     ..
 *     .. External Functions ..
       LOGICAL            LSAME, SISNAN
       EXTERNAL           LSAME, SISNAN
 *     ..
 *     .. External Subroutines ..
-      EXTERNAL           CLASSQ
+      EXTERNAL           CLASSQ, SCOMBSSQ
 *     ..
 *     .. Intrinsic Functions ..
       INTRINSIC          ABS, REAL, SQRT
@@ -223,31 +227,48 @@ REAL             FUNCTION CLANHE( NORM, UPLO, N, A, LDA, WORK )
       ELSE IF( ( LSAME( NORM, 'F' ) ) .OR. ( LSAME( NORM, 'E' ) ) ) THEN
 *
 *        Find normF(A).
+*        SSQ(1) is scale
+*        SSQ(2) is sum-of-squares
+*        For better accuracy, sum each column separately.
+*
+         SSQ( 1 ) = ZERO
+         SSQ( 2 ) = ONE
+*
+*        Sum off-diagonals
 *
-         SCALE = ZERO
-         SUM = ONE
          IF( LSAME( UPLO, 'U' ) ) THEN
             DO 110 J = 2, N
-               CALL CLASSQ( J-1, A( 1, J ), 1, SCALE, SUM )
+               COLSSQ( 1 ) = ZERO
+               COLSSQ( 2 ) = ONE
+               CALL CLASSQ( J-1, A( 1, J ), 1,
+     $                      COLSSQ( 1 ), COLSSQ( 2 ) )
+               CALL SCOMBSSQ( SSQ, COLSSQ )
   110       CONTINUE
          ELSE
             DO 120 J = 1, N - 1
-               CALL CLASSQ( N-J, A( J+1, J ), 1, SCALE, SUM )
+               COLSSQ( 1 ) = ZERO
+               COLSSQ( 2 ) = ONE
+               CALL CLASSQ( N-J, A( J+1, J ), 1,
+     $                      COLSSQ( 1 ), COLSSQ( 2 ) )
+               CALL SCOMBSSQ( SSQ, COLSSQ )
   120       CONTINUE
          END IF
-         SUM = 2*SUM
+         SSQ( 2 ) = 2*SSQ( 2 )
+*
+*        Sum diagonal
+*
          DO 130 I = 1, N
             IF( REAL( A( I, I ) ).NE.ZERO ) THEN
                ABSA = ABS( REAL( A( I, I ) ) )
-               IF( SCALE.LT.ABSA ) THEN
-                  SUM = ONE + SUM*( SCALE / ABSA )**2
-                  SCALE = ABSA
+               IF( SSQ( 1 ).LT.ABSA ) THEN
+                  SSQ( 2 ) = ONE + SSQ( 2 )*( SSQ( 1 ) / ABSA )**2
+                  SSQ( 1 ) = ABSA
                ELSE
-                  SUM = SUM + ( ABSA / SCALE )**2
+                  SSQ( 2 ) = SSQ( 2 ) + ( ABSA / SSQ( 1 ) )**2
                END IF
             END IF
   130    CONTINUE
-         VALUE = SCALE*SQRT( SUM )
+         VALUE = SSQ( 1 )*SQRT( SSQ( 2 ) )
       END IF
 *
       CLANHE = VALUE