LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
zhptrf.f
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00001 *> \brief \b ZHPTRF
00002 *
00003 *  =========== DOCUMENTATION ===========
00004 *
00005 * Online html documentation available at 
00006 *            http://www.netlib.org/lapack/explore-html/ 
00007 *
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00009 *> Download ZHPTRF + dependencies 
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00011 *> [TGZ]</a> 
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00015 *> [TXT]</a>
00016 *> \endhtmlonly 
00017 *
00018 *  Definition:
00019 *  ===========
00020 *
00021 *       SUBROUTINE ZHPTRF( UPLO, N, AP, IPIV, INFO )
00022 * 
00023 *       .. Scalar Arguments ..
00024 *       CHARACTER          UPLO
00025 *       INTEGER            INFO, N
00026 *       ..
00027 *       .. Array Arguments ..
00028 *       INTEGER            IPIV( * )
00029 *       COMPLEX*16         AP( * )
00030 *       ..
00031 *  
00032 *
00033 *> \par Purpose:
00034 *  =============
00035 *>
00036 *> \verbatim
00037 *>
00038 *> ZHPTRF computes the factorization of a complex Hermitian packed
00039 *> matrix A using the Bunch-Kaufman diagonal pivoting method:
00040 *>
00041 *>    A = U*D*U**H  or  A = L*D*L**H
00042 *>
00043 *> where U (or L) is a product of permutation and unit upper (lower)
00044 *> triangular matrices, and D is Hermitian and block diagonal with
00045 *> 1-by-1 and 2-by-2 diagonal blocks.
00046 *> \endverbatim
00047 *
00048 *  Arguments:
00049 *  ==========
00050 *
00051 *> \param[in] UPLO
00052 *> \verbatim
00053 *>          UPLO is CHARACTER*1
00054 *>          = 'U':  Upper triangle of A is stored;
00055 *>          = 'L':  Lower triangle of A is stored.
00056 *> \endverbatim
00057 *>
00058 *> \param[in] N
00059 *> \verbatim
00060 *>          N is INTEGER
00061 *>          The order of the matrix A.  N >= 0.
00062 *> \endverbatim
00063 *>
00064 *> \param[in,out] AP
00065 *> \verbatim
00066 *>          AP is COMPLEX*16 array, dimension (N*(N+1)/2)
00067 *>          On entry, the upper or lower triangle of the Hermitian matrix
00068 *>          A, packed columnwise in a linear array.  The j-th column of A
00069 *>          is stored in the array AP as follows:
00070 *>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
00071 *>          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
00072 *>
00073 *>          On exit, the block diagonal matrix D and the multipliers used
00074 *>          to obtain the factor U or L, stored as a packed triangular
00075 *>          matrix overwriting A (see below for further details).
00076 *> \endverbatim
00077 *>
00078 *> \param[out] IPIV
00079 *> \verbatim
00080 *>          IPIV is INTEGER array, dimension (N)
00081 *>          Details of the interchanges and the block structure of D.
00082 *>          If IPIV(k) > 0, then rows and columns k and IPIV(k) were
00083 *>          interchanged and D(k,k) is a 1-by-1 diagonal block.
00084 *>          If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
00085 *>          columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
00086 *>          is a 2-by-2 diagonal block.  If UPLO = 'L' and IPIV(k) =
00087 *>          IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
00088 *>          interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
00089 *> \endverbatim
00090 *>
00091 *> \param[out] INFO
00092 *> \verbatim
00093 *>          INFO is INTEGER
00094 *>          = 0: successful exit
00095 *>          < 0: if INFO = -i, the i-th argument had an illegal value
00096 *>          > 0: if INFO = i, D(i,i) is exactly zero.  The factorization
00097 *>               has been completed, but the block diagonal matrix D is
00098 *>               exactly singular, and division by zero will occur if it
00099 *>               is used to solve a system of equations.
00100 *> \endverbatim
00101 *
00102 *  Authors:
00103 *  ========
00104 *
00105 *> \author Univ. of Tennessee 
00106 *> \author Univ. of California Berkeley 
00107 *> \author Univ. of Colorado Denver 
00108 *> \author NAG Ltd. 
00109 *
00110 *> \date November 2011
00111 *
00112 *> \ingroup complex16OTHERcomputational
00113 *
00114 *> \par Further Details:
00115 *  =====================
00116 *>
00117 *> \verbatim
00118 *>
00119 *>  If UPLO = 'U', then A = U*D*U**H, where
00120 *>     U = P(n)*U(n)* ... *P(k)U(k)* ...,
00121 *>  i.e., U is a product of terms P(k)*U(k), where k decreases from n to
00122 *>  1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
00123 *>  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
00124 *>  defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
00125 *>  that if the diagonal block D(k) is of order s (s = 1 or 2), then
00126 *>
00127 *>             (   I    v    0   )   k-s
00128 *>     U(k) =  (   0    I    0   )   s
00129 *>             (   0    0    I   )   n-k
00130 *>                k-s   s   n-k
00131 *>
00132 *>  If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
00133 *>  If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
00134 *>  and A(k,k), and v overwrites A(1:k-2,k-1:k).
00135 *>
00136 *>  If UPLO = 'L', then A = L*D*L**H, where
00137 *>     L = P(1)*L(1)* ... *P(k)*L(k)* ...,
00138 *>  i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
00139 *>  n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
00140 *>  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
00141 *>  defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
00142 *>  that if the diagonal block D(k) is of order s (s = 1 or 2), then
00143 *>
00144 *>             (   I    0     0   )  k-1
00145 *>     L(k) =  (   0    I     0   )  s
00146 *>             (   0    v     I   )  n-k-s+1
00147 *>                k-1   s  n-k-s+1
00148 *>
00149 *>  If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
00150 *>  If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
00151 *>  and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
00152 *> \endverbatim
00153 *
00154 *> \par Contributors:
00155 *  ==================
00156 *>
00157 *>  J. Lewis, Boeing Computer Services Company
00158 *
00159 *  =====================================================================
00160       SUBROUTINE ZHPTRF( UPLO, N, AP, IPIV, INFO )
00161 *
00162 *  -- LAPACK computational routine (version 3.4.0) --
00163 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00164 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00165 *     November 2011
00166 *
00167 *     .. Scalar Arguments ..
00168       CHARACTER          UPLO
00169       INTEGER            INFO, N
00170 *     ..
00171 *     .. Array Arguments ..
00172       INTEGER            IPIV( * )
00173       COMPLEX*16         AP( * )
00174 *     ..
00175 *
00176 *  =====================================================================
00177 *
00178 *     .. Parameters ..
00179       DOUBLE PRECISION   ZERO, ONE
00180       PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
00181       DOUBLE PRECISION   EIGHT, SEVTEN
00182       PARAMETER          ( EIGHT = 8.0D+0, SEVTEN = 17.0D+0 )
00183 *     ..
00184 *     .. Local Scalars ..
00185       LOGICAL            UPPER
00186       INTEGER            I, IMAX, J, JMAX, K, KC, KK, KNC, KP, KPC,
00187      $                   KSTEP, KX, NPP
00188       DOUBLE PRECISION   ABSAKK, ALPHA, COLMAX, D, D11, D22, R1, ROWMAX,
00189      $                   TT
00190       COMPLEX*16         D12, D21, T, WK, WKM1, WKP1, ZDUM
00191 *     ..
00192 *     .. External Functions ..
00193       LOGICAL            LSAME
00194       INTEGER            IZAMAX
00195       DOUBLE PRECISION   DLAPY2
00196       EXTERNAL           LSAME, IZAMAX, DLAPY2
00197 *     ..
00198 *     .. External Subroutines ..
00199       EXTERNAL           XERBLA, ZDSCAL, ZHPR, ZSWAP
00200 *     ..
00201 *     .. Intrinsic Functions ..
00202       INTRINSIC          ABS, DBLE, DCMPLX, DCONJG, DIMAG, MAX, SQRT
00203 *     ..
00204 *     .. Statement Functions ..
00205       DOUBLE PRECISION   CABS1
00206 *     ..
00207 *     .. Statement Function definitions ..
00208       CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
00209 *     ..
00210 *     .. Executable Statements ..
00211 *
00212 *     Test the input parameters.
00213 *
00214       INFO = 0
00215       UPPER = LSAME( UPLO, 'U' )
00216       IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
00217          INFO = -1
00218       ELSE IF( N.LT.0 ) THEN
00219          INFO = -2
00220       END IF
00221       IF( INFO.NE.0 ) THEN
00222          CALL XERBLA( 'ZHPTRF', -INFO )
00223          RETURN
00224       END IF
00225 *
00226 *     Initialize ALPHA for use in choosing pivot block size.
00227 *
00228       ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT
00229 *
00230       IF( UPPER ) THEN
00231 *
00232 *        Factorize A as U*D*U**H using the upper triangle of A
00233 *
00234 *        K is the main loop index, decreasing from N to 1 in steps of
00235 *        1 or 2
00236 *
00237          K = N
00238          KC = ( N-1 )*N / 2 + 1
00239    10    CONTINUE
00240          KNC = KC
00241 *
00242 *        If K < 1, exit from loop
00243 *
00244          IF( K.LT.1 )
00245      $      GO TO 110
00246          KSTEP = 1
00247 *
00248 *        Determine rows and columns to be interchanged and whether
00249 *        a 1-by-1 or 2-by-2 pivot block will be used
00250 *
00251          ABSAKK = ABS( DBLE( AP( KC+K-1 ) ) )
00252 *
00253 *        IMAX is the row-index of the largest off-diagonal element in
00254 *        column K, and COLMAX is its absolute value
00255 *
00256          IF( K.GT.1 ) THEN
00257             IMAX = IZAMAX( K-1, AP( KC ), 1 )
00258             COLMAX = CABS1( AP( KC+IMAX-1 ) )
00259          ELSE
00260             COLMAX = ZERO
00261          END IF
00262 *
00263          IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
00264 *
00265 *           Column K is zero: set INFO and continue
00266 *
00267             IF( INFO.EQ.0 )
00268      $         INFO = K
00269             KP = K
00270             AP( KC+K-1 ) = DBLE( AP( KC+K-1 ) )
00271          ELSE
00272             IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
00273 *
00274 *              no interchange, use 1-by-1 pivot block
00275 *
00276                KP = K
00277             ELSE
00278 *
00279 *              JMAX is the column-index of the largest off-diagonal
00280 *              element in row IMAX, and ROWMAX is its absolute value
00281 *
00282                ROWMAX = ZERO
00283                JMAX = IMAX
00284                KX = IMAX*( IMAX+1 ) / 2 + IMAX
00285                DO 20 J = IMAX + 1, K
00286                   IF( CABS1( AP( KX ) ).GT.ROWMAX ) THEN
00287                      ROWMAX = CABS1( AP( KX ) )
00288                      JMAX = J
00289                   END IF
00290                   KX = KX + J
00291    20          CONTINUE
00292                KPC = ( IMAX-1 )*IMAX / 2 + 1
00293                IF( IMAX.GT.1 ) THEN
00294                   JMAX = IZAMAX( IMAX-1, AP( KPC ), 1 )
00295                   ROWMAX = MAX( ROWMAX, CABS1( AP( KPC+JMAX-1 ) ) )
00296                END IF
00297 *
00298                IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
00299 *
00300 *                 no interchange, use 1-by-1 pivot block
00301 *
00302                   KP = K
00303                ELSE IF( ABS( DBLE( AP( KPC+IMAX-1 ) ) ).GE.ALPHA*
00304      $                  ROWMAX ) THEN
00305 *
00306 *                 interchange rows and columns K and IMAX, use 1-by-1
00307 *                 pivot block
00308 *
00309                   KP = IMAX
00310                ELSE
00311 *
00312 *                 interchange rows and columns K-1 and IMAX, use 2-by-2
00313 *                 pivot block
00314 *
00315                   KP = IMAX
00316                   KSTEP = 2
00317                END IF
00318             END IF
00319 *
00320             KK = K - KSTEP + 1
00321             IF( KSTEP.EQ.2 )
00322      $         KNC = KNC - K + 1
00323             IF( KP.NE.KK ) THEN
00324 *
00325 *              Interchange rows and columns KK and KP in the leading
00326 *              submatrix A(1:k,1:k)
00327 *
00328                CALL ZSWAP( KP-1, AP( KNC ), 1, AP( KPC ), 1 )
00329                KX = KPC + KP - 1
00330                DO 30 J = KP + 1, KK - 1
00331                   KX = KX + J - 1
00332                   T = DCONJG( AP( KNC+J-1 ) )
00333                   AP( KNC+J-1 ) = DCONJG( AP( KX ) )
00334                   AP( KX ) = T
00335    30          CONTINUE
00336                AP( KX+KK-1 ) = DCONJG( AP( KX+KK-1 ) )
00337                R1 = DBLE( AP( KNC+KK-1 ) )
00338                AP( KNC+KK-1 ) = DBLE( AP( KPC+KP-1 ) )
00339                AP( KPC+KP-1 ) = R1
00340                IF( KSTEP.EQ.2 ) THEN
00341                   AP( KC+K-1 ) = DBLE( AP( KC+K-1 ) )
00342                   T = AP( KC+K-2 )
00343                   AP( KC+K-2 ) = AP( KC+KP-1 )
00344                   AP( KC+KP-1 ) = T
00345                END IF
00346             ELSE
00347                AP( KC+K-1 ) = DBLE( AP( KC+K-1 ) )
00348                IF( KSTEP.EQ.2 )
00349      $            AP( KC-1 ) = DBLE( AP( KC-1 ) )
00350             END IF
00351 *
00352 *           Update the leading submatrix
00353 *
00354             IF( KSTEP.EQ.1 ) THEN
00355 *
00356 *              1-by-1 pivot block D(k): column k now holds
00357 *
00358 *              W(k) = U(k)*D(k)
00359 *
00360 *              where U(k) is the k-th column of U
00361 *
00362 *              Perform a rank-1 update of A(1:k-1,1:k-1) as
00363 *
00364 *              A := A - U(k)*D(k)*U(k)**H = A - W(k)*1/D(k)*W(k)**H
00365 *
00366                R1 = ONE / DBLE( AP( KC+K-1 ) )
00367                CALL ZHPR( UPLO, K-1, -R1, AP( KC ), 1, AP )
00368 *
00369 *              Store U(k) in column k
00370 *
00371                CALL ZDSCAL( K-1, R1, AP( KC ), 1 )
00372             ELSE
00373 *
00374 *              2-by-2 pivot block D(k): columns k and k-1 now hold
00375 *
00376 *              ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
00377 *
00378 *              where U(k) and U(k-1) are the k-th and (k-1)-th columns
00379 *              of U
00380 *
00381 *              Perform a rank-2 update of A(1:k-2,1:k-2) as
00382 *
00383 *              A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )**H
00384 *                 = A - ( W(k-1) W(k) )*inv(D(k))*( W(k-1) W(k) )**H
00385 *
00386                IF( K.GT.2 ) THEN
00387 *
00388                   D = DLAPY2( DBLE( AP( K-1+( K-1 )*K / 2 ) ),
00389      $                DIMAG( AP( K-1+( K-1 )*K / 2 ) ) )
00390                   D22 = DBLE( AP( K-1+( K-2 )*( K-1 ) / 2 ) ) / D
00391                   D11 = DBLE( AP( K+( K-1 )*K / 2 ) ) / D
00392                   TT = ONE / ( D11*D22-ONE )
00393                   D12 = AP( K-1+( K-1 )*K / 2 ) / D
00394                   D = TT / D
00395 *
00396                   DO 50 J = K - 2, 1, -1
00397                      WKM1 = D*( D11*AP( J+( K-2 )*( K-1 ) / 2 )-
00398      $                      DCONJG( D12 )*AP( J+( K-1 )*K / 2 ) )
00399                      WK = D*( D22*AP( J+( K-1 )*K / 2 )-D12*
00400      $                    AP( J+( K-2 )*( K-1 ) / 2 ) )
00401                      DO 40 I = J, 1, -1
00402                         AP( I+( J-1 )*J / 2 ) = AP( I+( J-1 )*J / 2 ) -
00403      $                     AP( I+( K-1 )*K / 2 )*DCONJG( WK ) -
00404      $                     AP( I+( K-2 )*( K-1 ) / 2 )*DCONJG( WKM1 )
00405    40                CONTINUE
00406                      AP( J+( K-1 )*K / 2 ) = WK
00407                      AP( J+( K-2 )*( K-1 ) / 2 ) = WKM1
00408                      AP( J+( J-1 )*J / 2 ) = DCMPLX( DBLE( AP( J+( J-
00409      $                                       1 )*J / 2 ) ), 0.0D+0 )
00410    50             CONTINUE
00411 *
00412                END IF
00413 *
00414             END IF
00415          END IF
00416 *
00417 *        Store details of the interchanges in IPIV
00418 *
00419          IF( KSTEP.EQ.1 ) THEN
00420             IPIV( K ) = KP
00421          ELSE
00422             IPIV( K ) = -KP
00423             IPIV( K-1 ) = -KP
00424          END IF
00425 *
00426 *        Decrease K and return to the start of the main loop
00427 *
00428          K = K - KSTEP
00429          KC = KNC - K
00430          GO TO 10
00431 *
00432       ELSE
00433 *
00434 *        Factorize A as L*D*L**H using the lower triangle of A
00435 *
00436 *        K is the main loop index, increasing from 1 to N in steps of
00437 *        1 or 2
00438 *
00439          K = 1
00440          KC = 1
00441          NPP = N*( N+1 ) / 2
00442    60    CONTINUE
00443          KNC = KC
00444 *
00445 *        If K > N, exit from loop
00446 *
00447          IF( K.GT.N )
00448      $      GO TO 110
00449          KSTEP = 1
00450 *
00451 *        Determine rows and columns to be interchanged and whether
00452 *        a 1-by-1 or 2-by-2 pivot block will be used
00453 *
00454          ABSAKK = ABS( DBLE( AP( KC ) ) )
00455 *
00456 *        IMAX is the row-index of the largest off-diagonal element in
00457 *        column K, and COLMAX is its absolute value
00458 *
00459          IF( K.LT.N ) THEN
00460             IMAX = K + IZAMAX( N-K, AP( KC+1 ), 1 )
00461             COLMAX = CABS1( AP( KC+IMAX-K ) )
00462          ELSE
00463             COLMAX = ZERO
00464          END IF
00465 *
00466          IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
00467 *
00468 *           Column K is zero: set INFO and continue
00469 *
00470             IF( INFO.EQ.0 )
00471      $         INFO = K
00472             KP = K
00473             AP( KC ) = DBLE( AP( KC ) )
00474          ELSE
00475             IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
00476 *
00477 *              no interchange, use 1-by-1 pivot block
00478 *
00479                KP = K
00480             ELSE
00481 *
00482 *              JMAX is the column-index of the largest off-diagonal
00483 *              element in row IMAX, and ROWMAX is its absolute value
00484 *
00485                ROWMAX = ZERO
00486                KX = KC + IMAX - K
00487                DO 70 J = K, IMAX - 1
00488                   IF( CABS1( AP( KX ) ).GT.ROWMAX ) THEN
00489                      ROWMAX = CABS1( AP( KX ) )
00490                      JMAX = J
00491                   END IF
00492                   KX = KX + N - J
00493    70          CONTINUE
00494                KPC = NPP - ( N-IMAX+1 )*( N-IMAX+2 ) / 2 + 1
00495                IF( IMAX.LT.N ) THEN
00496                   JMAX = IMAX + IZAMAX( N-IMAX, AP( KPC+1 ), 1 )
00497                   ROWMAX = MAX( ROWMAX, CABS1( AP( KPC+JMAX-IMAX ) ) )
00498                END IF
00499 *
00500                IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
00501 *
00502 *                 no interchange, use 1-by-1 pivot block
00503 *
00504                   KP = K
00505                ELSE IF( ABS( DBLE( AP( KPC ) ) ).GE.ALPHA*ROWMAX ) THEN
00506 *
00507 *                 interchange rows and columns K and IMAX, use 1-by-1
00508 *                 pivot block
00509 *
00510                   KP = IMAX
00511                ELSE
00512 *
00513 *                 interchange rows and columns K+1 and IMAX, use 2-by-2
00514 *                 pivot block
00515 *
00516                   KP = IMAX
00517                   KSTEP = 2
00518                END IF
00519             END IF
00520 *
00521             KK = K + KSTEP - 1
00522             IF( KSTEP.EQ.2 )
00523      $         KNC = KNC + N - K + 1
00524             IF( KP.NE.KK ) THEN
00525 *
00526 *              Interchange rows and columns KK and KP in the trailing
00527 *              submatrix A(k:n,k:n)
00528 *
00529                IF( KP.LT.N )
00530      $            CALL ZSWAP( N-KP, AP( KNC+KP-KK+1 ), 1, AP( KPC+1 ),
00531      $                        1 )
00532                KX = KNC + KP - KK
00533                DO 80 J = KK + 1, KP - 1
00534                   KX = KX + N - J + 1
00535                   T = DCONJG( AP( KNC+J-KK ) )
00536                   AP( KNC+J-KK ) = DCONJG( AP( KX ) )
00537                   AP( KX ) = T
00538    80          CONTINUE
00539                AP( KNC+KP-KK ) = DCONJG( AP( KNC+KP-KK ) )
00540                R1 = DBLE( AP( KNC ) )
00541                AP( KNC ) = DBLE( AP( KPC ) )
00542                AP( KPC ) = R1
00543                IF( KSTEP.EQ.2 ) THEN
00544                   AP( KC ) = DBLE( AP( KC ) )
00545                   T = AP( KC+1 )
00546                   AP( KC+1 ) = AP( KC+KP-K )
00547                   AP( KC+KP-K ) = T
00548                END IF
00549             ELSE
00550                AP( KC ) = DBLE( AP( KC ) )
00551                IF( KSTEP.EQ.2 )
00552      $            AP( KNC ) = DBLE( AP( KNC ) )
00553             END IF
00554 *
00555 *           Update the trailing submatrix
00556 *
00557             IF( KSTEP.EQ.1 ) THEN
00558 *
00559 *              1-by-1 pivot block D(k): column k now holds
00560 *
00561 *              W(k) = L(k)*D(k)
00562 *
00563 *              where L(k) is the k-th column of L
00564 *
00565                IF( K.LT.N ) THEN
00566 *
00567 *                 Perform a rank-1 update of A(k+1:n,k+1:n) as
00568 *
00569 *                 A := A - L(k)*D(k)*L(k)**H = A - W(k)*(1/D(k))*W(k)**H
00570 *
00571                   R1 = ONE / DBLE( AP( KC ) )
00572                   CALL ZHPR( UPLO, N-K, -R1, AP( KC+1 ), 1,
00573      $                       AP( KC+N-K+1 ) )
00574 *
00575 *                 Store L(k) in column K
00576 *
00577                   CALL ZDSCAL( N-K, R1, AP( KC+1 ), 1 )
00578                END IF
00579             ELSE
00580 *
00581 *              2-by-2 pivot block D(k): columns K and K+1 now hold
00582 *
00583 *              ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k)
00584 *
00585 *              where L(k) and L(k+1) are the k-th and (k+1)-th columns
00586 *              of L
00587 *
00588                IF( K.LT.N-1 ) THEN
00589 *
00590 *                 Perform a rank-2 update of A(k+2:n,k+2:n) as
00591 *
00592 *                 A := A - ( L(k) L(k+1) )*D(k)*( L(k) L(k+1) )**H
00593 *                    = A - ( W(k) W(k+1) )*inv(D(k))*( W(k) W(k+1) )**H
00594 *
00595 *                 where L(k) and L(k+1) are the k-th and (k+1)-th
00596 *                 columns of L
00597 *
00598                   D = DLAPY2( DBLE( AP( K+1+( K-1 )*( 2*N-K ) / 2 ) ),
00599      $                DIMAG( AP( K+1+( K-1 )*( 2*N-K ) / 2 ) ) )
00600                   D11 = DBLE( AP( K+1+K*( 2*N-K-1 ) / 2 ) ) / D
00601                   D22 = DBLE( AP( K+( K-1 )*( 2*N-K ) / 2 ) ) / D
00602                   TT = ONE / ( D11*D22-ONE )
00603                   D21 = AP( K+1+( K-1 )*( 2*N-K ) / 2 ) / D
00604                   D = TT / D
00605 *
00606                   DO 100 J = K + 2, N
00607                      WK = D*( D11*AP( J+( K-1 )*( 2*N-K ) / 2 )-D21*
00608      $                    AP( J+K*( 2*N-K-1 ) / 2 ) )
00609                      WKP1 = D*( D22*AP( J+K*( 2*N-K-1 ) / 2 )-
00610      $                      DCONJG( D21 )*AP( J+( K-1 )*( 2*N-K ) /
00611      $                      2 ) )
00612                      DO 90 I = J, N
00613                         AP( I+( J-1 )*( 2*N-J ) / 2 ) = AP( I+( J-1 )*
00614      $                     ( 2*N-J ) / 2 ) - AP( I+( K-1 )*( 2*N-K ) /
00615      $                     2 )*DCONJG( WK ) - AP( I+K*( 2*N-K-1 ) / 2 )*
00616      $                     DCONJG( WKP1 )
00617    90                CONTINUE
00618                      AP( J+( K-1 )*( 2*N-K ) / 2 ) = WK
00619                      AP( J+K*( 2*N-K-1 ) / 2 ) = WKP1
00620                      AP( J+( J-1 )*( 2*N-J ) / 2 )
00621      $                  = DCMPLX( DBLE( AP( J+( J-1 )*( 2*N-J ) / 2 ) ),
00622      $                  0.0D+0 )
00623   100             CONTINUE
00624                END IF
00625             END IF
00626          END IF
00627 *
00628 *        Store details of the interchanges in IPIV
00629 *
00630          IF( KSTEP.EQ.1 ) THEN
00631             IPIV( K ) = KP
00632          ELSE
00633             IPIV( K ) = -KP
00634             IPIV( K+1 ) = -KP
00635          END IF
00636 *
00637 *        Increase K and return to the start of the main loop
00638 *
00639          K = K + KSTEP
00640          KC = KNC + N - K + 2
00641          GO TO 60
00642 *
00643       END IF
00644 *
00645   110 CONTINUE
00646       RETURN
00647 *
00648 *     End of ZHPTRF
00649 *
00650       END
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