--- rpl/lapack/lapack/zlansb.f 2010/08/07 13:22:39 1.5
+++ rpl/lapack/lapack/zlansb.f 2020/05/21 21:46:08 1.18
@@ -1,11 +1,141 @@
+*> \brief \b ZLANSB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a symmetric band matrix.
+*
+* =========== DOCUMENTATION ===========
+*
+* Online html documentation available at
+* http://www.netlib.org/lapack/explore-html/
+*
+*> \htmlonly
+*> Download ZLANSB + dependencies
+*>
+*> [TGZ]
+*>
+*> [ZIP]
+*>
+*> [TXT]
+*> \endhtmlonly
+*
+* Definition:
+* ===========
+*
+* DOUBLE PRECISION FUNCTION ZLANSB( NORM, UPLO, N, K, AB, LDAB,
+* WORK )
+*
+* .. Scalar Arguments ..
+* CHARACTER NORM, UPLO
+* INTEGER K, LDAB, N
+* ..
+* .. Array Arguments ..
+* DOUBLE PRECISION WORK( * )
+* COMPLEX*16 AB( LDAB, * )
+* ..
+*
+*
+*> \par Purpose:
+* =============
+*>
+*> \verbatim
+*>
+*> ZLANSB returns the value of the one norm, or the Frobenius norm, or
+*> the infinity norm, or the element of largest absolute value of an
+*> n by n symmetric band matrix A, with k super-diagonals.
+*> \endverbatim
+*>
+*> \return ZLANSB
+*> \verbatim
+*>
+*> ZLANSB = ( max(abs(A(i,j))), NORM = 'M' or 'm'
+*> (
+*> ( norm1(A), NORM = '1', 'O' or 'o'
+*> (
+*> ( normI(A), NORM = 'I' or 'i'
+*> (
+*> ( normF(A), NORM = 'F', 'f', 'E' or 'e'
+*>
+*> where norm1 denotes the one norm of a matrix (maximum column sum),
+*> normI denotes the infinity norm of a matrix (maximum row sum) and
+*> normF denotes the Frobenius norm of a matrix (square root of sum of
+*> squares). Note that max(abs(A(i,j))) is not a consistent matrix norm.
+*> \endverbatim
+*
+* Arguments:
+* ==========
+*
+*> \param[in] NORM
+*> \verbatim
+*> NORM is CHARACTER*1
+*> Specifies the value to be returned in ZLANSB as described
+*> above.
+*> \endverbatim
+*>
+*> \param[in] UPLO
+*> \verbatim
+*> UPLO is CHARACTER*1
+*> Specifies whether the upper or lower triangular part of the
+*> band matrix A is supplied.
+*> = 'U': Upper triangular part is supplied
+*> = 'L': Lower triangular part is supplied
+*> \endverbatim
+*>
+*> \param[in] N
+*> \verbatim
+*> N is INTEGER
+*> The order of the matrix A. N >= 0. When N = 0, ZLANSB is
+*> set to zero.
+*> \endverbatim
+*>
+*> \param[in] K
+*> \verbatim
+*> K is INTEGER
+*> The number of super-diagonals or sub-diagonals of the
+*> band matrix A. K >= 0.
+*> \endverbatim
+*>
+*> \param[in] AB
+*> \verbatim
+*> AB is COMPLEX*16 array, dimension (LDAB,N)
+*> The upper or lower triangle of the symmetric band matrix A,
+*> stored in the first K+1 rows of AB. The j-th column of A is
+*> stored in the j-th column of the array AB as follows:
+*> if UPLO = 'U', AB(k+1+i-j,j) = A(i,j) for max(1,j-k)<=i<=j;
+*> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+k).
+*> \endverbatim
+*>
+*> \param[in] LDAB
+*> \verbatim
+*> LDAB is INTEGER
+*> The leading dimension of the array AB. LDAB >= K+1.
+*> \endverbatim
+*>
+*> \param[out] WORK
+*> \verbatim
+*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
+*> where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
+*> WORK is not referenced.
+*> \endverbatim
+*
+* Authors:
+* ========
+*
+*> \author Univ. of Tennessee
+*> \author Univ. of California Berkeley
+*> \author Univ. of Colorado Denver
+*> \author NAG Ltd.
+*
+*> \date December 2016
+*
+*> \ingroup complex16OTHERauxiliary
+*
+* =====================================================================
DOUBLE PRECISION FUNCTION ZLANSB( NORM, UPLO, N, K, AB, LDAB,
$ WORK )
*
-* -- LAPACK auxiliary routine (version 3.2) --
+* -- LAPACK auxiliary routine (version 3.7.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
-* November 2006
+* December 2016
*
+ IMPLICIT NONE
* .. Scalar Arguments ..
CHARACTER NORM, UPLO
INTEGER K, LDAB, N
@@ -15,66 +145,6 @@
COMPLEX*16 AB( LDAB, * )
* ..
*
-* Purpose
-* =======
-*
-* ZLANSB returns the value of the one norm, or the Frobenius norm, or
-* the infinity norm, or the element of largest absolute value of an
-* n by n symmetric band matrix A, with k super-diagonals.
-*
-* Description
-* ===========
-*
-* ZLANSB returns the value
-*
-* ZLANSB = ( max(abs(A(i,j))), NORM = 'M' or 'm'
-* (
-* ( norm1(A), NORM = '1', 'O' or 'o'
-* (
-* ( normI(A), NORM = 'I' or 'i'
-* (
-* ( normF(A), NORM = 'F', 'f', 'E' or 'e'
-*
-* where norm1 denotes the one norm of a matrix (maximum column sum),
-* normI denotes the infinity norm of a matrix (maximum row sum) and
-* normF denotes the Frobenius norm of a matrix (square root of sum of
-* squares). Note that max(abs(A(i,j))) is not a consistent matrix norm.
-*
-* Arguments
-* =========
-*
-* NORM (input) CHARACTER*1
-* Specifies the value to be returned in ZLANSB as described
-* above.
-*
-* UPLO (input) CHARACTER*1
-* Specifies whether the upper or lower triangular part of the
-* band matrix A is supplied.
-* = 'U': Upper triangular part is supplied
-* = 'L': Lower triangular part is supplied
-*
-* N (input) INTEGER
-* The order of the matrix A. N >= 0. When N = 0, ZLANSB is
-* set to zero.
-*
-* K (input) INTEGER
-* The number of super-diagonals or sub-diagonals of the
-* band matrix A. K >= 0.
-*
-* AB (input) COMPLEX*16 array, dimension (LDAB,N)
-* The upper or lower triangle of the symmetric band matrix A,
-* stored in the first K+1 rows of AB. The j-th column of A is
-* stored in the j-th column of the array AB as follows:
-* if UPLO = 'U', AB(k+1+i-j,j) = A(i,j) for max(1,j-k)<=i<=j;
-* if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+k).
-*
-* LDAB (input) INTEGER
-* The leading dimension of the array AB. LDAB >= K+1.
-*
-* WORK (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
-* where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
-* WORK is not referenced.
-*
* =====================================================================
*
* .. Parameters ..
@@ -83,14 +153,17 @@
* ..
* .. Local Scalars ..
INTEGER I, J, L
- DOUBLE PRECISION ABSA, SCALE, SUM, VALUE
+ DOUBLE PRECISION ABSA, SUM, VALUE
+* ..
+* .. Local Arrays ..
+ DOUBLE PRECISION SSQ( 2 ), COLSSQ( 2 )
* ..
* .. External Functions ..
- LOGICAL LSAME
- EXTERNAL LSAME
+ LOGICAL LSAME, DISNAN
+ EXTERNAL LSAME, DISNAN
* ..
* .. External Subroutines ..
- EXTERNAL ZLASSQ
+ EXTERNAL ZLASSQ, DCOMBSSQ
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN, SQRT
@@ -107,13 +180,15 @@
IF( LSAME( UPLO, 'U' ) ) THEN
DO 20 J = 1, N
DO 10 I = MAX( K+2-J, 1 ), K + 1
- VALUE = MAX( VALUE, ABS( AB( I, J ) ) )
+ SUM = ABS( AB( I, J ) )
+ IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
10 CONTINUE
20 CONTINUE
ELSE
DO 40 J = 1, N
DO 30 I = 1, MIN( N+1-J, K+1 )
- VALUE = MAX( VALUE, ABS( AB( I, J ) ) )
+ SUM = ABS( AB( I, J ) )
+ IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
30 CONTINUE
40 CONTINUE
END IF
@@ -135,7 +210,8 @@
WORK( J ) = SUM + ABS( AB( K+1, J ) )
60 CONTINUE
DO 70 I = 1, N
- VALUE = MAX( VALUE, WORK( I ) )
+ SUM = WORK( I )
+ IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
70 CONTINUE
ELSE
DO 80 I = 1, N
@@ -149,35 +225,53 @@
SUM = SUM + ABSA
WORK( I ) = WORK( I ) + ABSA
90 CONTINUE
- VALUE = MAX( VALUE, SUM )
+ IF( VALUE .LT. SUM .OR. DISNAN( SUM ) ) VALUE = SUM
100 CONTINUE
END IF
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 ZLASSQ( MIN( J-1, K ), AB( MAX( K+2-J, 1 ), J ),
- $ 1, SCALE, SUM )
+ $ 1, COLSSQ( 1 ), COLSSQ( 2 ) )
+ CALL DCOMBSSQ( SSQ, COLSSQ )
110 CONTINUE
L = K + 1
ELSE
DO 120 J = 1, N - 1
- CALL ZLASSQ( MIN( N-J, K ), AB( 2, J ), 1, SCALE,
- $ SUM )
+ COLSSQ( 1 ) = ZERO
+ COLSSQ( 2 ) = ONE
+ CALL ZLASSQ( MIN( N-J, K ), AB( 2, J ), 1,
+ $ COLSSQ( 1 ), COLSSQ( 2 ) )
+ CALL DCOMBSSQ( SSQ, COLSSQ )
120 CONTINUE
L = 1
END IF
- SUM = 2*SUM
+ SSQ( 2 ) = 2*SSQ( 2 )
ELSE
L = 1
END IF
- CALL ZLASSQ( N, AB( L, 1 ), LDAB, SCALE, SUM )
- VALUE = SCALE*SQRT( SUM )
+*
+* Sum diagonal
+*
+ COLSSQ( 1 ) = ZERO
+ COLSSQ( 2 ) = ONE
+ CALL ZLASSQ( N, AB( L, 1 ), LDAB, COLSSQ( 1 ), COLSSQ( 2 ) )
+ CALL DCOMBSSQ( SSQ, COLSSQ )
+ VALUE = SSQ( 1 )*SQRT( SSQ( 2 ) )
END IF
*
ZLANSB = VALUE