LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
zsptrf.f
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00001 *> \brief \b ZSPTRF
00002 *
00003 *  =========== DOCUMENTATION ===========
00004 *
00005 * Online html documentation available at 
00006 *            http://www.netlib.org/lapack/explore-html/ 
00007 *
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00011 *> [TGZ]</a> 
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00015 *> [TXT]</a>
00016 *> \endhtmlonly 
00017 *
00018 *  Definition:
00019 *  ===========
00020 *
00021 *       SUBROUTINE ZSPTRF( 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 *> ZSPTRF computes the factorization of a complex symmetric matrix A
00039 *> stored in packed format using the Bunch-Kaufman diagonal pivoting
00040 *> method:
00041 *>
00042 *>    A = U*D*U**T  or  A = L*D*L**T
00043 *>
00044 *> where U (or L) is a product of permutation and unit upper (lower)
00045 *> triangular matrices, and D is symmetric and block diagonal with
00046 *> 1-by-1 and 2-by-2 diagonal blocks.
00047 *> \endverbatim
00048 *
00049 *  Arguments:
00050 *  ==========
00051 *
00052 *> \param[in] UPLO
00053 *> \verbatim
00054 *>          UPLO is CHARACTER*1
00055 *>          = 'U':  Upper triangle of A is stored;
00056 *>          = 'L':  Lower triangle of A is stored.
00057 *> \endverbatim
00058 *>
00059 *> \param[in] N
00060 *> \verbatim
00061 *>          N is INTEGER
00062 *>          The order of the matrix A.  N >= 0.
00063 *> \endverbatim
00064 *>
00065 *> \param[in,out] AP
00066 *> \verbatim
00067 *>          AP is COMPLEX*16 array, dimension (N*(N+1)/2)
00068 *>          On entry, the upper or lower triangle of the symmetric matrix
00069 *>          A, packed columnwise in a linear array.  The j-th column of A
00070 *>          is stored in the array AP as follows:
00071 *>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
00072 *>          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
00073 *>
00074 *>          On exit, the block diagonal matrix D and the multipliers used
00075 *>          to obtain the factor U or L, stored as a packed triangular
00076 *>          matrix overwriting A (see below for further details).
00077 *> \endverbatim
00078 *>
00079 *> \param[out] IPIV
00080 *> \verbatim
00081 *>          IPIV is INTEGER array, dimension (N)
00082 *>          Details of the interchanges and the block structure of D.
00083 *>          If IPIV(k) > 0, then rows and columns k and IPIV(k) were
00084 *>          interchanged and D(k,k) is a 1-by-1 diagonal block.
00085 *>          If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
00086 *>          columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
00087 *>          is a 2-by-2 diagonal block.  If UPLO = 'L' and IPIV(k) =
00088 *>          IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
00089 *>          interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
00090 *> \endverbatim
00091 *>
00092 *> \param[out] INFO
00093 *> \verbatim
00094 *>          INFO is INTEGER
00095 *>          = 0: successful exit
00096 *>          < 0: if INFO = -i, the i-th argument had an illegal value
00097 *>          > 0: if INFO = i, D(i,i) is exactly zero.  The factorization
00098 *>               has been completed, but the block diagonal matrix D is
00099 *>               exactly singular, and division by zero will occur if it
00100 *>               is used to solve a system of equations.
00101 *> \endverbatim
00102 *
00103 *  Authors:
00104 *  ========
00105 *
00106 *> \author Univ. of Tennessee 
00107 *> \author Univ. of California Berkeley 
00108 *> \author Univ. of Colorado Denver 
00109 *> \author NAG Ltd. 
00110 *
00111 *> \date November 2011
00112 *
00113 *> \ingroup complex16OTHERcomputational
00114 *
00115 *> \par Further Details:
00116 *  =====================
00117 *>
00118 *> \verbatim
00119 *>
00120 *>  5-96 - Based on modifications by J. Lewis, Boeing Computer Services
00121 *>         Company
00122 *>
00123 *>  If UPLO = 'U', then A = U*D*U**T, where
00124 *>     U = P(n)*U(n)* ... *P(k)U(k)* ...,
00125 *>  i.e., U is a product of terms P(k)*U(k), where k decreases from n to
00126 *>  1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
00127 *>  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
00128 *>  defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
00129 *>  that if the diagonal block D(k) is of order s (s = 1 or 2), then
00130 *>
00131 *>             (   I    v    0   )   k-s
00132 *>     U(k) =  (   0    I    0   )   s
00133 *>             (   0    0    I   )   n-k
00134 *>                k-s   s   n-k
00135 *>
00136 *>  If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
00137 *>  If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
00138 *>  and A(k,k), and v overwrites A(1:k-2,k-1:k).
00139 *>
00140 *>  If UPLO = 'L', then A = L*D*L**T, where
00141 *>     L = P(1)*L(1)* ... *P(k)*L(k)* ...,
00142 *>  i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
00143 *>  n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
00144 *>  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
00145 *>  defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
00146 *>  that if the diagonal block D(k) is of order s (s = 1 or 2), then
00147 *>
00148 *>             (   I    0     0   )  k-1
00149 *>     L(k) =  (   0    I     0   )  s
00150 *>             (   0    v     I   )  n-k-s+1
00151 *>                k-1   s  n-k-s+1
00152 *>
00153 *>  If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
00154 *>  If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
00155 *>  and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
00156 *> \endverbatim
00157 *>
00158 *  =====================================================================
00159       SUBROUTINE ZSPTRF( UPLO, N, AP, IPIV, INFO )
00160 *
00161 *  -- LAPACK computational routine (version 3.4.0) --
00162 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00163 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00164 *     November 2011
00165 *
00166 *     .. Scalar Arguments ..
00167       CHARACTER          UPLO
00168       INTEGER            INFO, N
00169 *     ..
00170 *     .. Array Arguments ..
00171       INTEGER            IPIV( * )
00172       COMPLEX*16         AP( * )
00173 *     ..
00174 *
00175 *  =====================================================================
00176 *
00177 *     .. Parameters ..
00178       DOUBLE PRECISION   ZERO, ONE
00179       PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
00180       DOUBLE PRECISION   EIGHT, SEVTEN
00181       PARAMETER          ( EIGHT = 8.0D+0, SEVTEN = 17.0D+0 )
00182       COMPLEX*16         CONE
00183       PARAMETER          ( CONE = ( 1.0D+0, 0.0D+0 ) )
00184 *     ..
00185 *     .. Local Scalars ..
00186       LOGICAL            UPPER
00187       INTEGER            I, IMAX, J, JMAX, K, KC, KK, KNC, KP, KPC,
00188      $                   KSTEP, KX, NPP
00189       DOUBLE PRECISION   ABSAKK, ALPHA, COLMAX, ROWMAX
00190       COMPLEX*16         D11, D12, D21, D22, R1, T, WK, WKM1, WKP1, ZDUM
00191 *     ..
00192 *     .. External Functions ..
00193       LOGICAL            LSAME
00194       INTEGER            IZAMAX
00195       EXTERNAL           LSAME, IZAMAX
00196 *     ..
00197 *     .. External Subroutines ..
00198       EXTERNAL           XERBLA, ZSCAL, ZSPR, ZSWAP
00199 *     ..
00200 *     .. Intrinsic Functions ..
00201       INTRINSIC          ABS, DBLE, DIMAG, MAX, SQRT
00202 *     ..
00203 *     .. Statement Functions ..
00204       DOUBLE PRECISION   CABS1
00205 *     ..
00206 *     .. Statement Function definitions ..
00207       CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
00208 *     ..
00209 *     .. Executable Statements ..
00210 *
00211 *     Test the input parameters.
00212 *
00213       INFO = 0
00214       UPPER = LSAME( UPLO, 'U' )
00215       IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
00216          INFO = -1
00217       ELSE IF( N.LT.0 ) THEN
00218          INFO = -2
00219       END IF
00220       IF( INFO.NE.0 ) THEN
00221          CALL XERBLA( 'ZSPTRF', -INFO )
00222          RETURN
00223       END IF
00224 *
00225 *     Initialize ALPHA for use in choosing pivot block size.
00226 *
00227       ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT
00228 *
00229       IF( UPPER ) THEN
00230 *
00231 *        Factorize A as U*D*U**T using the upper triangle of A
00232 *
00233 *        K is the main loop index, decreasing from N to 1 in steps of
00234 *        1 or 2
00235 *
00236          K = N
00237          KC = ( N-1 )*N / 2 + 1
00238    10    CONTINUE
00239          KNC = KC
00240 *
00241 *        If K < 1, exit from loop
00242 *
00243          IF( K.LT.1 )
00244      $      GO TO 110
00245          KSTEP = 1
00246 *
00247 *        Determine rows and columns to be interchanged and whether
00248 *        a 1-by-1 or 2-by-2 pivot block will be used
00249 *
00250          ABSAKK = CABS1( AP( KC+K-1 ) )
00251 *
00252 *        IMAX is the row-index of the largest off-diagonal element in
00253 *        column K, and COLMAX is its absolute value
00254 *
00255          IF( K.GT.1 ) THEN
00256             IMAX = IZAMAX( K-1, AP( KC ), 1 )
00257             COLMAX = CABS1( AP( KC+IMAX-1 ) )
00258          ELSE
00259             COLMAX = ZERO
00260          END IF
00261 *
00262          IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
00263 *
00264 *           Column K is zero: set INFO and continue
00265 *
00266             IF( INFO.EQ.0 )
00267      $         INFO = K
00268             KP = K
00269          ELSE
00270             IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
00271 *
00272 *              no interchange, use 1-by-1 pivot block
00273 *
00274                KP = K
00275             ELSE
00276 *
00277                ROWMAX = ZERO
00278                JMAX = IMAX
00279                KX = IMAX*( IMAX+1 ) / 2 + IMAX
00280                DO 20 J = IMAX + 1, K
00281                   IF( CABS1( AP( KX ) ).GT.ROWMAX ) THEN
00282                      ROWMAX = CABS1( AP( KX ) )
00283                      JMAX = J
00284                   END IF
00285                   KX = KX + J
00286    20          CONTINUE
00287                KPC = ( IMAX-1 )*IMAX / 2 + 1
00288                IF( IMAX.GT.1 ) THEN
00289                   JMAX = IZAMAX( IMAX-1, AP( KPC ), 1 )
00290                   ROWMAX = MAX( ROWMAX, CABS1( AP( KPC+JMAX-1 ) ) )
00291                END IF
00292 *
00293                IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
00294 *
00295 *                 no interchange, use 1-by-1 pivot block
00296 *
00297                   KP = K
00298                ELSE IF( CABS1( AP( KPC+IMAX-1 ) ).GE.ALPHA*ROWMAX ) THEN
00299 *
00300 *                 interchange rows and columns K and IMAX, use 1-by-1
00301 *                 pivot block
00302 *
00303                   KP = IMAX
00304                ELSE
00305 *
00306 *                 interchange rows and columns K-1 and IMAX, use 2-by-2
00307 *                 pivot block
00308 *
00309                   KP = IMAX
00310                   KSTEP = 2
00311                END IF
00312             END IF
00313 *
00314             KK = K - KSTEP + 1
00315             IF( KSTEP.EQ.2 )
00316      $         KNC = KNC - K + 1
00317             IF( KP.NE.KK ) THEN
00318 *
00319 *              Interchange rows and columns KK and KP in the leading
00320 *              submatrix A(1:k,1:k)
00321 *
00322                CALL ZSWAP( KP-1, AP( KNC ), 1, AP( KPC ), 1 )
00323                KX = KPC + KP - 1
00324                DO 30 J = KP + 1, KK - 1
00325                   KX = KX + J - 1
00326                   T = AP( KNC+J-1 )
00327                   AP( KNC+J-1 ) = AP( KX )
00328                   AP( KX ) = T
00329    30          CONTINUE
00330                T = AP( KNC+KK-1 )
00331                AP( KNC+KK-1 ) = AP( KPC+KP-1 )
00332                AP( KPC+KP-1 ) = T
00333                IF( KSTEP.EQ.2 ) THEN
00334                   T = AP( KC+K-2 )
00335                   AP( KC+K-2 ) = AP( KC+KP-1 )
00336                   AP( KC+KP-1 ) = T
00337                END IF
00338             END IF
00339 *
00340 *           Update the leading submatrix
00341 *
00342             IF( KSTEP.EQ.1 ) THEN
00343 *
00344 *              1-by-1 pivot block D(k): column k now holds
00345 *
00346 *              W(k) = U(k)*D(k)
00347 *
00348 *              where U(k) is the k-th column of U
00349 *
00350 *              Perform a rank-1 update of A(1:k-1,1:k-1) as
00351 *
00352 *              A := A - U(k)*D(k)*U(k)**T = A - W(k)*1/D(k)*W(k)**T
00353 *
00354                R1 = CONE / AP( KC+K-1 )
00355                CALL ZSPR( UPLO, K-1, -R1, AP( KC ), 1, AP )
00356 *
00357 *              Store U(k) in column k
00358 *
00359                CALL ZSCAL( K-1, R1, AP( KC ), 1 )
00360             ELSE
00361 *
00362 *              2-by-2 pivot block D(k): columns k and k-1 now hold
00363 *
00364 *              ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
00365 *
00366 *              where U(k) and U(k-1) are the k-th and (k-1)-th columns
00367 *              of U
00368 *
00369 *              Perform a rank-2 update of A(1:k-2,1:k-2) as
00370 *
00371 *              A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )**T
00372 *                 = A - ( W(k-1) W(k) )*inv(D(k))*( W(k-1) W(k) )**T
00373 *
00374                IF( K.GT.2 ) THEN
00375 *
00376                   D12 = AP( K-1+( K-1 )*K / 2 )
00377                   D22 = AP( K-1+( K-2 )*( K-1 ) / 2 ) / D12
00378                   D11 = AP( K+( K-1 )*K / 2 ) / D12
00379                   T = CONE / ( D11*D22-CONE )
00380                   D12 = T / D12
00381 *
00382                   DO 50 J = K - 2, 1, -1
00383                      WKM1 = D12*( D11*AP( J+( K-2 )*( K-1 ) / 2 )-
00384      $                      AP( J+( K-1 )*K / 2 ) )
00385                      WK = D12*( D22*AP( J+( K-1 )*K / 2 )-
00386      $                    AP( J+( K-2 )*( K-1 ) / 2 ) )
00387                      DO 40 I = J, 1, -1
00388                         AP( I+( J-1 )*J / 2 ) = AP( I+( J-1 )*J / 2 ) -
00389      $                     AP( I+( K-1 )*K / 2 )*WK -
00390      $                     AP( I+( K-2 )*( K-1 ) / 2 )*WKM1
00391    40                CONTINUE
00392                      AP( J+( K-1 )*K / 2 ) = WK
00393                      AP( J+( K-2 )*( K-1 ) / 2 ) = WKM1
00394    50             CONTINUE
00395 *
00396                END IF
00397             END IF
00398          END IF
00399 *
00400 *        Store details of the interchanges in IPIV
00401 *
00402          IF( KSTEP.EQ.1 ) THEN
00403             IPIV( K ) = KP
00404          ELSE
00405             IPIV( K ) = -KP
00406             IPIV( K-1 ) = -KP
00407          END IF
00408 *
00409 *        Decrease K and return to the start of the main loop
00410 *
00411          K = K - KSTEP
00412          KC = KNC - K
00413          GO TO 10
00414 *
00415       ELSE
00416 *
00417 *        Factorize A as L*D*L**T using the lower triangle of A
00418 *
00419 *        K is the main loop index, increasing from 1 to N in steps of
00420 *        1 or 2
00421 *
00422          K = 1
00423          KC = 1
00424          NPP = N*( N+1 ) / 2
00425    60    CONTINUE
00426          KNC = KC
00427 *
00428 *        If K > N, exit from loop
00429 *
00430          IF( K.GT.N )
00431      $      GO TO 110
00432          KSTEP = 1
00433 *
00434 *        Determine rows and columns to be interchanged and whether
00435 *        a 1-by-1 or 2-by-2 pivot block will be used
00436 *
00437          ABSAKK = CABS1( AP( KC ) )
00438 *
00439 *        IMAX is the row-index of the largest off-diagonal element in
00440 *        column K, and COLMAX is its absolute value
00441 *
00442          IF( K.LT.N ) THEN
00443             IMAX = K + IZAMAX( N-K, AP( KC+1 ), 1 )
00444             COLMAX = CABS1( AP( KC+IMAX-K ) )
00445          ELSE
00446             COLMAX = ZERO
00447          END IF
00448 *
00449          IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
00450 *
00451 *           Column K is zero: set INFO and continue
00452 *
00453             IF( INFO.EQ.0 )
00454      $         INFO = K
00455             KP = K
00456          ELSE
00457             IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
00458 *
00459 *              no interchange, use 1-by-1 pivot block
00460 *
00461                KP = K
00462             ELSE
00463 *
00464 *              JMAX is the column-index of the largest off-diagonal
00465 *              element in row IMAX, and ROWMAX is its absolute value
00466 *
00467                ROWMAX = ZERO
00468                KX = KC + IMAX - K
00469                DO 70 J = K, IMAX - 1
00470                   IF( CABS1( AP( KX ) ).GT.ROWMAX ) THEN
00471                      ROWMAX = CABS1( AP( KX ) )
00472                      JMAX = J
00473                   END IF
00474                   KX = KX + N - J
00475    70          CONTINUE
00476                KPC = NPP - ( N-IMAX+1 )*( N-IMAX+2 ) / 2 + 1
00477                IF( IMAX.LT.N ) THEN
00478                   JMAX = IMAX + IZAMAX( N-IMAX, AP( KPC+1 ), 1 )
00479                   ROWMAX = MAX( ROWMAX, CABS1( AP( KPC+JMAX-IMAX ) ) )
00480                END IF
00481 *
00482                IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
00483 *
00484 *                 no interchange, use 1-by-1 pivot block
00485 *
00486                   KP = K
00487                ELSE IF( CABS1( AP( KPC ) ).GE.ALPHA*ROWMAX ) THEN
00488 *
00489 *                 interchange rows and columns K and IMAX, use 1-by-1
00490 *                 pivot block
00491 *
00492                   KP = IMAX
00493                ELSE
00494 *
00495 *                 interchange rows and columns K+1 and IMAX, use 2-by-2
00496 *                 pivot block
00497 *
00498                   KP = IMAX
00499                   KSTEP = 2
00500                END IF
00501             END IF
00502 *
00503             KK = K + KSTEP - 1
00504             IF( KSTEP.EQ.2 )
00505      $         KNC = KNC + N - K + 1
00506             IF( KP.NE.KK ) THEN
00507 *
00508 *              Interchange rows and columns KK and KP in the trailing
00509 *              submatrix A(k:n,k:n)
00510 *
00511                IF( KP.LT.N )
00512      $            CALL ZSWAP( N-KP, AP( KNC+KP-KK+1 ), 1, AP( KPC+1 ),
00513      $                        1 )
00514                KX = KNC + KP - KK
00515                DO 80 J = KK + 1, KP - 1
00516                   KX = KX + N - J + 1
00517                   T = AP( KNC+J-KK )
00518                   AP( KNC+J-KK ) = AP( KX )
00519                   AP( KX ) = T
00520    80          CONTINUE
00521                T = AP( KNC )
00522                AP( KNC ) = AP( KPC )
00523                AP( KPC ) = T
00524                IF( KSTEP.EQ.2 ) THEN
00525                   T = AP( KC+1 )
00526                   AP( KC+1 ) = AP( KC+KP-K )
00527                   AP( KC+KP-K ) = T
00528                END IF
00529             END IF
00530 *
00531 *           Update the trailing submatrix
00532 *
00533             IF( KSTEP.EQ.1 ) THEN
00534 *
00535 *              1-by-1 pivot block D(k): column k now holds
00536 *
00537 *              W(k) = L(k)*D(k)
00538 *
00539 *              where L(k) is the k-th column of L
00540 *
00541                IF( K.LT.N ) THEN
00542 *
00543 *                 Perform a rank-1 update of A(k+1:n,k+1:n) as
00544 *
00545 *                 A := A - L(k)*D(k)*L(k)**T = A - W(k)*(1/D(k))*W(k)**T
00546 *
00547                   R1 = CONE / AP( KC )
00548                   CALL ZSPR( UPLO, N-K, -R1, AP( KC+1 ), 1,
00549      $                       AP( KC+N-K+1 ) )
00550 *
00551 *                 Store L(k) in column K
00552 *
00553                   CALL ZSCAL( N-K, R1, AP( KC+1 ), 1 )
00554                END IF
00555             ELSE
00556 *
00557 *              2-by-2 pivot block D(k): columns K and K+1 now hold
00558 *
00559 *              ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k)
00560 *
00561 *              where L(k) and L(k+1) are the k-th and (k+1)-th columns
00562 *              of L
00563 *
00564                IF( K.LT.N-1 ) THEN
00565 *
00566 *                 Perform a rank-2 update of A(k+2:n,k+2:n) as
00567 *
00568 *                 A := A - ( L(k) L(k+1) )*D(k)*( L(k) L(k+1) )**T
00569 *                    = A - ( W(k) W(k+1) )*inv(D(k))*( W(k) W(k+1) )**T
00570 *
00571 *                 where L(k) and L(k+1) are the k-th and (k+1)-th
00572 *                 columns of L
00573 *
00574                   D21 = AP( K+1+( K-1 )*( 2*N-K ) / 2 )
00575                   D11 = AP( K+1+K*( 2*N-K-1 ) / 2 ) / D21
00576                   D22 = AP( K+( K-1 )*( 2*N-K ) / 2 ) / D21
00577                   T = CONE / ( D11*D22-CONE )
00578                   D21 = T / D21
00579 *
00580                   DO 100 J = K + 2, N
00581                      WK = D21*( D11*AP( J+( K-1 )*( 2*N-K ) / 2 )-
00582      $                    AP( J+K*( 2*N-K-1 ) / 2 ) )
00583                      WKP1 = D21*( D22*AP( J+K*( 2*N-K-1 ) / 2 )-
00584      $                      AP( J+( K-1 )*( 2*N-K ) / 2 ) )
00585                      DO 90 I = J, N
00586                         AP( I+( J-1 )*( 2*N-J ) / 2 ) = AP( I+( J-1 )*
00587      $                     ( 2*N-J ) / 2 ) - AP( I+( K-1 )*( 2*N-K ) /
00588      $                     2 )*WK - AP( I+K*( 2*N-K-1 ) / 2 )*WKP1
00589    90                CONTINUE
00590                      AP( J+( K-1 )*( 2*N-K ) / 2 ) = WK
00591                      AP( J+K*( 2*N-K-1 ) / 2 ) = WKP1
00592   100             CONTINUE
00593                END IF
00594             END IF
00595          END IF
00596 *
00597 *        Store details of the interchanges in IPIV
00598 *
00599          IF( KSTEP.EQ.1 ) THEN
00600             IPIV( K ) = KP
00601          ELSE
00602             IPIV( K ) = -KP
00603             IPIV( K+1 ) = -KP
00604          END IF
00605 *
00606 *        Increase K and return to the start of the main loop
00607 *
00608          K = K + KSTEP
00609          KC = KNC + N - K + 2
00610          GO TO 60
00611 *
00612       END IF
00613 *
00614   110 CONTINUE
00615       RETURN
00616 *
00617 *     End of ZSPTRF
00618 *
00619       END
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