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version 1.5, 2010/08/07 13:22:35 version 1.14, 2016/08/27 15:34:52
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   *> \brief \b ZHPTRF
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at 
   *            http://www.netlib.org/lapack/explore-html/ 
   *
   *> \htmlonly
   *> Download ZHPTRF + dependencies 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zhptrf.f"> 
   *> [TGZ]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zhptrf.f"> 
   *> [ZIP]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zhptrf.f"> 
   *> [TXT]</a>
   *> \endhtmlonly 
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE ZHPTRF( UPLO, N, AP, IPIV, INFO )
   * 
   *       .. Scalar Arguments ..
   *       CHARACTER          UPLO
   *       INTEGER            INFO, N
   *       ..
   *       .. Array Arguments ..
   *       INTEGER            IPIV( * )
   *       COMPLEX*16         AP( * )
   *       ..
   *  
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> ZHPTRF computes the factorization of a complex Hermitian packed
   *> matrix A using the Bunch-Kaufman diagonal pivoting method:
   *>
   *>    A = U*D*U**H  or  A = L*D*L**H
   *>
   *> where U (or L) is a product of permutation and unit upper (lower)
   *> triangular matrices, and D is Hermitian and block diagonal with
   *> 1-by-1 and 2-by-2 diagonal blocks.
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \param[in] UPLO
   *> \verbatim
   *>          UPLO is CHARACTER*1
   *>          = 'U':  Upper triangle of A is stored;
   *>          = 'L':  Lower triangle of A is stored.
   *> \endverbatim
   *>
   *> \param[in] N
   *> \verbatim
   *>          N is INTEGER
   *>          The order of the matrix A.  N >= 0.
   *> \endverbatim
   *>
   *> \param[in,out] AP
   *> \verbatim
   *>          AP is COMPLEX*16 array, dimension (N*(N+1)/2)
   *>          On entry, the upper or lower triangle of the Hermitian matrix
   *>          A, packed columnwise in a linear array.  The j-th column of A
   *>          is stored in the array AP as follows:
   *>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
   *>          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
   *>
   *>          On exit, the block diagonal matrix D and the multipliers used
   *>          to obtain the factor U or L, stored as a packed triangular
   *>          matrix overwriting A (see below for further details).
   *> \endverbatim
   *>
   *> \param[out] IPIV
   *> \verbatim
   *>          IPIV is INTEGER array, dimension (N)
   *>          Details of the interchanges and the block structure of D.
   *>          If IPIV(k) > 0, then rows and columns k and IPIV(k) were
   *>          interchanged and D(k,k) is a 1-by-1 diagonal block.
   *>          If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
   *>          columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
   *>          is a 2-by-2 diagonal block.  If UPLO = 'L' and IPIV(k) =
   *>          IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
   *>          interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
   *> \endverbatim
   *>
   *> \param[out] INFO
   *> \verbatim
   *>          INFO is INTEGER
   *>          = 0: successful exit
   *>          < 0: if INFO = -i, the i-th argument had an illegal value
   *>          > 0: if INFO = i, D(i,i) is exactly zero.  The factorization
   *>               has been completed, but the block diagonal matrix D is
   *>               exactly singular, and division by zero will occur if it
   *>               is used to solve a system of equations.
   *> \endverbatim
   *
   *  Authors:
   *  ========
   *
   *> \author Univ. of Tennessee 
   *> \author Univ. of California Berkeley 
   *> \author Univ. of Colorado Denver 
   *> \author NAG Ltd. 
   *
   *> \date November 2011
   *
   *> \ingroup complex16OTHERcomputational
   *
   *> \par Further Details:
   *  =====================
   *>
   *> \verbatim
   *>
   *>  If UPLO = 'U', then A = U*D*U**H, where
   *>     U = P(n)*U(n)* ... *P(k)U(k)* ...,
   *>  i.e., U is a product of terms P(k)*U(k), where k decreases from n to
   *>  1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
   *>  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
   *>  defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
   *>  that if the diagonal block D(k) is of order s (s = 1 or 2), then
   *>
   *>             (   I    v    0   )   k-s
   *>     U(k) =  (   0    I    0   )   s
   *>             (   0    0    I   )   n-k
   *>                k-s   s   n-k
   *>
   *>  If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
   *>  If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
   *>  and A(k,k), and v overwrites A(1:k-2,k-1:k).
   *>
   *>  If UPLO = 'L', then A = L*D*L**H, where
   *>     L = P(1)*L(1)* ... *P(k)*L(k)* ...,
   *>  i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
   *>  n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
   *>  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
   *>  defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
   *>  that if the diagonal block D(k) is of order s (s = 1 or 2), then
   *>
   *>             (   I    0     0   )  k-1
   *>     L(k) =  (   0    I     0   )  s
   *>             (   0    v     I   )  n-k-s+1
   *>                k-1   s  n-k-s+1
   *>
   *>  If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
   *>  If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
   *>  and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
   *> \endverbatim
   *
   *> \par Contributors:
   *  ==================
   *>
   *>  J. Lewis, Boeing Computer Services Company
   *
   *  =====================================================================
       SUBROUTINE ZHPTRF( UPLO, N, AP, IPIV, INFO )        SUBROUTINE ZHPTRF( UPLO, N, AP, IPIV, INFO )
 *  *
 *  -- LAPACK routine (version 3.2) --  *  -- LAPACK computational routine (version 3.4.0) --
 *  -- 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  *     November 2011
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       CHARACTER          UPLO        CHARACTER          UPLO
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       COMPLEX*16         AP( * )        COMPLEX*16         AP( * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  ZHPTRF computes the factorization of a complex Hermitian packed  
 *  matrix A using the Bunch-Kaufman diagonal pivoting method:  
 *  
 *     A = U*D*U**H  or  A = L*D*L**H  
 *  
 *  where U (or L) is a product of permutation and unit upper (lower)  
 *  triangular matrices, and D is Hermitian and block diagonal with  
 *  1-by-1 and 2-by-2 diagonal blocks.  
 *  
 *  Arguments  
 *  =========  
 *  
 *  UPLO    (input) CHARACTER*1  
 *          = 'U':  Upper triangle of A is stored;  
 *          = 'L':  Lower triangle of A is stored.  
 *  
 *  N       (input) INTEGER  
 *          The order of the matrix A.  N >= 0.  
 *  
 *  AP      (input/output) COMPLEX*16 array, dimension (N*(N+1)/2)  
 *          On entry, the upper or lower triangle of the Hermitian matrix  
 *          A, packed columnwise in a linear array.  The j-th column of A  
 *          is stored in the array AP as follows:  
 *          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;  
 *          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.  
 *  
 *          On exit, the block diagonal matrix D and the multipliers used  
 *          to obtain the factor U or L, stored as a packed triangular  
 *          matrix overwriting A (see below for further details).  
 *  
 *  IPIV    (output) INTEGER array, dimension (N)  
 *          Details of the interchanges and the block structure of D.  
 *          If IPIV(k) > 0, then rows and columns k and IPIV(k) were  
 *          interchanged and D(k,k) is a 1-by-1 diagonal block.  
 *          If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and  
 *          columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)  
 *          is a 2-by-2 diagonal block.  If UPLO = 'L' and IPIV(k) =  
 *          IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were  
 *          interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.  
 *  
 *  INFO    (output) INTEGER  
 *          = 0: successful exit  
 *          < 0: if INFO = -i, the i-th argument had an illegal value  
 *          > 0: if INFO = i, D(i,i) is exactly zero.  The factorization  
 *               has been completed, but the block diagonal matrix D is  
 *               exactly singular, and division by zero will occur if it  
 *               is used to solve a system of equations.  
 *  
 *  Further Details  
 *  ===============  
 *  
 *  5-96 - Based on modifications by J. Lewis, Boeing Computer Services  
 *         Company  
 *  
 *  If UPLO = 'U', then A = U*D*U', where  
 *     U = P(n)*U(n)* ... *P(k)U(k)* ...,  
 *  i.e., U is a product of terms P(k)*U(k), where k decreases from n to  
 *  1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1  
 *  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as  
 *  defined by IPIV(k), and U(k) is a unit upper triangular matrix, such  
 *  that if the diagonal block D(k) is of order s (s = 1 or 2), then  
 *  
 *             (   I    v    0   )   k-s  
 *     U(k) =  (   0    I    0   )   s  
 *             (   0    0    I   )   n-k  
 *                k-s   s   n-k  
 *  
 *  If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).  
 *  If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),  
 *  and A(k,k), and v overwrites A(1:k-2,k-1:k).  
 *  
 *  If UPLO = 'L', then A = L*D*L', where  
 *     L = P(1)*L(1)* ... *P(k)*L(k)* ...,  
 *  i.e., L is a product of terms P(k)*L(k), where k increases from 1 to  
 *  n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1  
 *  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as  
 *  defined by IPIV(k), and L(k) is a unit lower triangular matrix, such  
 *  that if the diagonal block D(k) is of order s (s = 1 or 2), then  
 *  
 *             (   I    0     0   )  k-1  
 *     L(k) =  (   0    I     0   )  s  
 *             (   0    v     I   )  n-k-s+1  
 *                k-1   s  n-k-s+1  
 *  
 *  If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).  
 *  If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),  
 *  and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Parameters ..  *     .. Parameters ..
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 *  *
       IF( UPPER ) THEN        IF( UPPER ) THEN
 *  *
 *        Factorize A as U*D*U' using the upper triangle of A  *        Factorize A as U*D*U**H using the upper triangle of A
 *  *
 *        K is the main loop index, decreasing from N to 1 in steps of  *        K is the main loop index, decreasing from N to 1 in steps of
 *        1 or 2  *        1 or 2
Line 293 Line 361
 *  *
 *              Perform a rank-1 update of A(1:k-1,1:k-1) as  *              Perform a rank-1 update of A(1:k-1,1:k-1) as
 *  *
 *              A := A - U(k)*D(k)*U(k)' = A - W(k)*1/D(k)*W(k)'  *              A := A - U(k)*D(k)*U(k)**H = A - W(k)*1/D(k)*W(k)**H
 *  *
                R1 = ONE / DBLE( AP( KC+K-1 ) )                 R1 = ONE / DBLE( AP( KC+K-1 ) )
                CALL ZHPR( UPLO, K-1, -R1, AP( KC ), 1, AP )                 CALL ZHPR( UPLO, K-1, -R1, AP( KC ), 1, AP )
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 *  *
 *              Perform a rank-2 update of A(1:k-2,1:k-2) as  *              Perform a rank-2 update of A(1:k-2,1:k-2) as
 *  *
 *              A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )'  *              A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )**H
 *                 = A - ( W(k-1) W(k) )*inv(D(k))*( W(k-1) W(k) )'  *                 = A - ( W(k-1) W(k) )*inv(D(k))*( W(k-1) W(k) )**H
 *  *
                IF( K.GT.2 ) THEN                 IF( K.GT.2 ) THEN
 *  *
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 *  *
       ELSE        ELSE
 *  *
 *        Factorize A as L*D*L' using the lower triangle of A  *        Factorize A as L*D*L**H using the lower triangle of A
 *  *
 *        K is the main loop index, increasing from 1 to N in steps of  *        K is the main loop index, increasing from 1 to N in steps of
 *        1 or 2  *        1 or 2
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 *  *
 *                 Perform a rank-1 update of A(k+1:n,k+1:n) as  *                 Perform a rank-1 update of A(k+1:n,k+1:n) as
 *  *
 *                 A := A - L(k)*D(k)*L(k)' = A - W(k)*(1/D(k))*W(k)'  *                 A := A - L(k)*D(k)*L(k)**H = A - W(k)*(1/D(k))*W(k)**H
 *  *
                   R1 = ONE / DBLE( AP( KC ) )                    R1 = ONE / DBLE( AP( KC ) )
                   CALL ZHPR( UPLO, N-K, -R1, AP( KC+1 ), 1,                    CALL ZHPR( UPLO, N-K, -R1, AP( KC+1 ), 1,
Line 521 Line 589
 *  *
 *                 Perform a rank-2 update of A(k+2:n,k+2:n) as  *                 Perform a rank-2 update of A(k+2:n,k+2:n) as
 *  *
 *                 A := A - ( L(k) L(k+1) )*D(k)*( L(k) L(k+1) )'  *                 A := A - ( L(k) L(k+1) )*D(k)*( L(k) L(k+1) )**H
 *                    = A - ( W(k) W(k+1) )*inv(D(k))*( W(k) W(k+1) )'  *                    = A - ( W(k) W(k+1) )*inv(D(k))*( W(k) W(k+1) )**H
 *  *
 *                 where L(k) and L(k+1) are the k-th and (k+1)-th  *                 where L(k) and L(k+1) are the k-th and (k+1)-th
 *                 columns of L  *                 columns of L
Removed from v.1.5  
changed lines
  Added in v.1.14

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