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