LAPACK 3.3.0

zsytf2.f

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00001       SUBROUTINE ZSYTF2( UPLO, N, A, LDA, IPIV, INFO )
00002 *
00003 *  -- LAPACK routine (version 3.2) --
00004 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00005 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00006 *     November 2006
00007 *
00008 *     .. Scalar Arguments ..
00009       CHARACTER          UPLO
00010       INTEGER            INFO, LDA, N
00011 *     ..
00012 *     .. Array Arguments ..
00013       INTEGER            IPIV( * )
00014       COMPLEX*16         A( LDA, * )
00015 *     ..
00016 *
00017 *  Purpose
00018 *  =======
00019 *
00020 *  ZSYTF2 computes the factorization of a complex symmetric matrix A
00021 *  using the Bunch-Kaufman diagonal pivoting method:
00022 *
00023 *     A = U*D*U'  or  A = L*D*L'
00024 *
00025 *  where U (or L) is a product of permutation and unit upper (lower)
00026 *  triangular matrices, U' is the transpose of U, and D is symmetric and
00027 *  block diagonal with 1-by-1 and 2-by-2 diagonal blocks.
00028 *
00029 *  This is the unblocked version of the algorithm, calling Level 2 BLAS.
00030 *
00031 *  Arguments
00032 *  =========
00033 *
00034 *  UPLO    (input) CHARACTER*1
00035 *          Specifies whether the upper or lower triangular part of the
00036 *          symmetric matrix A is stored:
00037 *          = 'U':  Upper triangular
00038 *          = 'L':  Lower triangular
00039 *
00040 *  N       (input) INTEGER
00041 *          The order of the matrix A.  N >= 0.
00042 *
00043 *  A       (input/output) COMPLEX*16 array, dimension (LDA,N)
00044 *          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
00045 *          n-by-n upper triangular part of A contains the upper
00046 *          triangular part of the matrix A, and the strictly lower
00047 *          triangular part of A is not referenced.  If UPLO = 'L', the
00048 *          leading n-by-n lower triangular part of A contains the lower
00049 *          triangular part of the matrix A, and the strictly upper
00050 *          triangular part of A is not referenced.
00051 *
00052 *          On exit, the block diagonal matrix D and the multipliers used
00053 *          to obtain the factor U or L (see below for further details).
00054 *
00055 *  LDA     (input) INTEGER
00056 *          The leading dimension of the array A.  LDA >= max(1,N).
00057 *
00058 *  IPIV    (output) INTEGER array, dimension (N)
00059 *          Details of the interchanges and the block structure of D.
00060 *          If IPIV(k) > 0, then rows and columns k and IPIV(k) were
00061 *          interchanged and D(k,k) is a 1-by-1 diagonal block.
00062 *          If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
00063 *          columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
00064 *          is a 2-by-2 diagonal block.  If UPLO = 'L' and IPIV(k) =
00065 *          IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
00066 *          interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
00067 *
00068 *  INFO    (output) INTEGER
00069 *          = 0: successful exit
00070 *          < 0: if INFO = -k, the k-th argument had an illegal value
00071 *          > 0: if INFO = k, D(k,k) is exactly zero.  The factorization
00072 *               has been completed, but the block diagonal matrix D is
00073 *               exactly singular, and division by zero will occur if it
00074 *               is used to solve a system of equations.
00075 *
00076 *  Further Details
00077 *  ===============
00078 *
00079 *  09-29-06 - patch from
00080 *    Bobby Cheng, MathWorks
00081 *
00082 *    Replace l.209 and l.377
00083 *         IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
00084 *    by
00085 *         IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. DISNAN(ABSAKK) ) THEN
00086 *
00087 *  1-96 - Based on modifications by J. Lewis, Boeing Computer Services
00088 *         Company
00089 *
00090 *  If UPLO = 'U', then A = U*D*U', where
00091 *     U = P(n)*U(n)* ... *P(k)U(k)* ...,
00092 *  i.e., U is a product of terms P(k)*U(k), where k decreases from n to
00093 *  1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
00094 *  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
00095 *  defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
00096 *  that if the diagonal block D(k) is of order s (s = 1 or 2), then
00097 *
00098 *             (   I    v    0   )   k-s
00099 *     U(k) =  (   0    I    0   )   s
00100 *             (   0    0    I   )   n-k
00101 *                k-s   s   n-k
00102 *
00103 *  If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
00104 *  If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
00105 *  and A(k,k), and v overwrites A(1:k-2,k-1:k).
00106 *
00107 *  If UPLO = 'L', then A = L*D*L', where
00108 *     L = P(1)*L(1)* ... *P(k)*L(k)* ...,
00109 *  i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
00110 *  n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
00111 *  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
00112 *  defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
00113 *  that if the diagonal block D(k) is of order s (s = 1 or 2), then
00114 *
00115 *             (   I    0     0   )  k-1
00116 *     L(k) =  (   0    I     0   )  s
00117 *             (   0    v     I   )  n-k-s+1
00118 *                k-1   s  n-k-s+1
00119 *
00120 *  If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
00121 *  If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
00122 *  and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
00123 *
00124 *  =====================================================================
00125 *
00126 *     .. Parameters ..
00127       DOUBLE PRECISION   ZERO, ONE
00128       PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
00129       DOUBLE PRECISION   EIGHT, SEVTEN
00130       PARAMETER          ( EIGHT = 8.0D+0, SEVTEN = 17.0D+0 )
00131       COMPLEX*16         CONE
00132       PARAMETER          ( CONE = ( 1.0D+0, 0.0D+0 ) )
00133 *     ..
00134 *     .. Local Scalars ..
00135       LOGICAL            UPPER
00136       INTEGER            I, IMAX, J, JMAX, K, KK, KP, KSTEP
00137       DOUBLE PRECISION   ABSAKK, ALPHA, COLMAX, ROWMAX
00138       COMPLEX*16         D11, D12, D21, D22, R1, T, WK, WKM1, WKP1, Z
00139 *     ..
00140 *     .. External Functions ..
00141       LOGICAL            DISNAN, LSAME
00142       INTEGER            IZAMAX
00143       EXTERNAL           DISNAN, LSAME, IZAMAX
00144 *     ..
00145 *     .. External Subroutines ..
00146       EXTERNAL           XERBLA, ZSCAL, ZSWAP, ZSYR
00147 *     ..
00148 *     .. Intrinsic Functions ..
00149       INTRINSIC          ABS, DBLE, DIMAG, MAX, SQRT
00150 *     ..
00151 *     .. Statement Functions ..
00152       DOUBLE PRECISION   CABS1
00153 *     ..
00154 *     .. Statement Function definitions ..
00155       CABS1( Z ) = ABS( DBLE( Z ) ) + ABS( DIMAG( Z ) )
00156 *     ..
00157 *     .. Executable Statements ..
00158 *
00159 *     Test the input parameters.
00160 *
00161       INFO = 0
00162       UPPER = LSAME( UPLO, 'U' )
00163       IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
00164          INFO = -1
00165       ELSE IF( N.LT.0 ) THEN
00166          INFO = -2
00167       ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
00168          INFO = -4
00169       END IF
00170       IF( INFO.NE.0 ) THEN
00171          CALL XERBLA( 'ZSYTF2', -INFO )
00172          RETURN
00173       END IF
00174 *
00175 *     Initialize ALPHA for use in choosing pivot block size.
00176 *
00177       ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT
00178 *
00179       IF( UPPER ) THEN
00180 *
00181 *        Factorize A as U*D*U' using the upper triangle of A
00182 *
00183 *        K is the main loop index, decreasing from N to 1 in steps of
00184 *        1 or 2
00185 *
00186          K = N
00187    10    CONTINUE
00188 *
00189 *        If K < 1, exit from loop
00190 *
00191          IF( K.LT.1 )
00192      $      GO TO 70
00193          KSTEP = 1
00194 *
00195 *        Determine rows and columns to be interchanged and whether
00196 *        a 1-by-1 or 2-by-2 pivot block will be used
00197 *
00198          ABSAKK = CABS1( A( K, K ) )
00199 *
00200 *        IMAX is the row-index of the largest off-diagonal element in
00201 *        column K, and COLMAX is its absolute value
00202 *
00203          IF( K.GT.1 ) THEN
00204             IMAX = IZAMAX( K-1, A( 1, K ), 1 )
00205             COLMAX = CABS1( A( IMAX, K ) )
00206          ELSE
00207             COLMAX = ZERO
00208          END IF
00209 *
00210          IF( MAX( ABSAKK, COLMAX ).EQ.ZERO .OR. DISNAN(ABSAKK) ) THEN
00211 *
00212 *           Column K is zero or contains a NaN: set INFO and continue
00213 *
00214             IF( INFO.EQ.0 )
00215      $         INFO = K
00216             KP = K
00217          ELSE
00218             IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
00219 *
00220 *              no interchange, use 1-by-1 pivot block
00221 *
00222                KP = K
00223             ELSE
00224 *
00225 *              JMAX is the column-index of the largest off-diagonal
00226 *              element in row IMAX, and ROWMAX is its absolute value
00227 *
00228                JMAX = IMAX + IZAMAX( K-IMAX, A( IMAX, IMAX+1 ), LDA )
00229                ROWMAX = CABS1( A( IMAX, JMAX ) )
00230                IF( IMAX.GT.1 ) THEN
00231                   JMAX = IZAMAX( IMAX-1, A( 1, IMAX ), 1 )
00232                   ROWMAX = MAX( ROWMAX, CABS1( A( JMAX, IMAX ) ) )
00233                END IF
00234 *
00235                IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
00236 *
00237 *                 no interchange, use 1-by-1 pivot block
00238 *
00239                   KP = K
00240                ELSE IF( CABS1( A( IMAX, IMAX ) ).GE.ALPHA*ROWMAX ) THEN
00241 *
00242 *                 interchange rows and columns K and IMAX, use 1-by-1
00243 *                 pivot block
00244 *
00245                   KP = IMAX
00246                ELSE
00247 *
00248 *                 interchange rows and columns K-1 and IMAX, use 2-by-2
00249 *                 pivot block
00250 *
00251                   KP = IMAX
00252                   KSTEP = 2
00253                END IF
00254             END IF
00255 *
00256             KK = K - KSTEP + 1
00257             IF( KP.NE.KK ) THEN
00258 *
00259 *              Interchange rows and columns KK and KP in the leading
00260 *              submatrix A(1:k,1:k)
00261 *
00262                CALL ZSWAP( KP-1, A( 1, KK ), 1, A( 1, KP ), 1 )
00263                CALL ZSWAP( KK-KP-1, A( KP+1, KK ), 1, A( KP, KP+1 ),
00264      $                     LDA )
00265                T = A( KK, KK )
00266                A( KK, KK ) = A( KP, KP )
00267                A( KP, KP ) = T
00268                IF( KSTEP.EQ.2 ) THEN
00269                   T = A( K-1, K )
00270                   A( K-1, K ) = A( KP, K )
00271                   A( KP, K ) = T
00272                END IF
00273             END IF
00274 *
00275 *           Update the leading submatrix
00276 *
00277             IF( KSTEP.EQ.1 ) THEN
00278 *
00279 *              1-by-1 pivot block D(k): column k now holds
00280 *
00281 *              W(k) = U(k)*D(k)
00282 *
00283 *              where U(k) is the k-th column of U
00284 *
00285 *              Perform a rank-1 update of A(1:k-1,1:k-1) as
00286 *
00287 *              A := A - U(k)*D(k)*U(k)' = A - W(k)*1/D(k)*W(k)'
00288 *
00289                R1 = CONE / A( K, K )
00290                CALL ZSYR( UPLO, K-1, -R1, A( 1, K ), 1, A, LDA )
00291 *
00292 *              Store U(k) in column k
00293 *
00294                CALL ZSCAL( K-1, R1, A( 1, K ), 1 )
00295             ELSE
00296 *
00297 *              2-by-2 pivot block D(k): columns k and k-1 now hold
00298 *
00299 *              ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
00300 *
00301 *              where U(k) and U(k-1) are the k-th and (k-1)-th columns
00302 *              of U
00303 *
00304 *              Perform a rank-2 update of A(1:k-2,1:k-2) as
00305 *
00306 *              A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )'
00307 *                 = A - ( W(k-1) W(k) )*inv(D(k))*( W(k-1) W(k) )'
00308 *
00309                IF( K.GT.2 ) THEN
00310 *
00311                   D12 = A( K-1, K )
00312                   D22 = A( K-1, K-1 ) / D12
00313                   D11 = A( K, K ) / D12
00314                   T = CONE / ( D11*D22-CONE )
00315                   D12 = T / D12
00316 *
00317                   DO 30 J = K - 2, 1, -1
00318                      WKM1 = D12*( D11*A( J, K-1 )-A( J, K ) )
00319                      WK = D12*( D22*A( J, K )-A( J, K-1 ) )
00320                      DO 20 I = J, 1, -1
00321                         A( I, J ) = A( I, J ) - A( I, K )*WK -
00322      $                              A( I, K-1 )*WKM1
00323    20                CONTINUE
00324                      A( J, K ) = WK
00325                      A( J, K-1 ) = WKM1
00326    30             CONTINUE
00327 *
00328                END IF
00329 *
00330             END IF
00331          END IF
00332 *
00333 *        Store details of the interchanges in IPIV
00334 *
00335          IF( KSTEP.EQ.1 ) THEN
00336             IPIV( K ) = KP
00337          ELSE
00338             IPIV( K ) = -KP
00339             IPIV( K-1 ) = -KP
00340          END IF
00341 *
00342 *        Decrease K and return to the start of the main loop
00343 *
00344          K = K - KSTEP
00345          GO TO 10
00346 *
00347       ELSE
00348 *
00349 *        Factorize A as L*D*L' using the lower triangle of A
00350 *
00351 *        K is the main loop index, increasing from 1 to N in steps of
00352 *        1 or 2
00353 *
00354          K = 1
00355    40    CONTINUE
00356 *
00357 *        If K > N, exit from loop
00358 *
00359          IF( K.GT.N )
00360      $      GO TO 70
00361          KSTEP = 1
00362 *
00363 *        Determine rows and columns to be interchanged and whether
00364 *        a 1-by-1 or 2-by-2 pivot block will be used
00365 *
00366          ABSAKK = CABS1( A( K, K ) )
00367 *
00368 *        IMAX is the row-index of the largest off-diagonal element in
00369 *        column K, and COLMAX is its absolute value
00370 *
00371          IF( K.LT.N ) THEN
00372             IMAX = K + IZAMAX( N-K, A( K+1, K ), 1 )
00373             COLMAX = CABS1( A( IMAX, K ) )
00374          ELSE
00375             COLMAX = ZERO
00376          END IF
00377 *
00378          IF( MAX( ABSAKK, COLMAX ).EQ.ZERO .OR. DISNAN(ABSAKK) ) THEN
00379 *
00380 *           Column K is zero or contains a NaN: set INFO and continue
00381 *
00382             IF( INFO.EQ.0 )
00383      $         INFO = K
00384             KP = K
00385          ELSE
00386             IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
00387 *
00388 *              no interchange, use 1-by-1 pivot block
00389 *
00390                KP = K
00391             ELSE
00392 *
00393 *              JMAX is the column-index of the largest off-diagonal
00394 *              element in row IMAX, and ROWMAX is its absolute value
00395 *
00396                JMAX = K - 1 + IZAMAX( IMAX-K, A( IMAX, K ), LDA )
00397                ROWMAX = CABS1( A( IMAX, JMAX ) )
00398                IF( IMAX.LT.N ) THEN
00399                   JMAX = IMAX + IZAMAX( N-IMAX, A( IMAX+1, IMAX ), 1 )
00400                   ROWMAX = MAX( ROWMAX, CABS1( A( JMAX, IMAX ) ) )
00401                END IF
00402 *
00403                IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
00404 *
00405 *                 no interchange, use 1-by-1 pivot block
00406 *
00407                   KP = K
00408                ELSE IF( CABS1( A( IMAX, IMAX ) ).GE.ALPHA*ROWMAX ) THEN
00409 *
00410 *                 interchange rows and columns K and IMAX, use 1-by-1
00411 *                 pivot block
00412 *
00413                   KP = IMAX
00414                ELSE
00415 *
00416 *                 interchange rows and columns K+1 and IMAX, use 2-by-2
00417 *                 pivot block
00418 *
00419                   KP = IMAX
00420                   KSTEP = 2
00421                END IF
00422             END IF
00423 *
00424             KK = K + KSTEP - 1
00425             IF( KP.NE.KK ) THEN
00426 *
00427 *              Interchange rows and columns KK and KP in the trailing
00428 *              submatrix A(k:n,k:n)
00429 *
00430                IF( KP.LT.N )
00431      $            CALL ZSWAP( N-KP, A( KP+1, KK ), 1, A( KP+1, KP ), 1 )
00432                CALL ZSWAP( KP-KK-1, A( KK+1, KK ), 1, A( KP, KK+1 ),
00433      $                     LDA )
00434                T = A( KK, KK )
00435                A( KK, KK ) = A( KP, KP )
00436                A( KP, KP ) = T
00437                IF( KSTEP.EQ.2 ) THEN
00438                   T = A( K+1, K )
00439                   A( K+1, K ) = A( KP, K )
00440                   A( KP, K ) = T
00441                END IF
00442             END IF
00443 *
00444 *           Update the trailing submatrix
00445 *
00446             IF( KSTEP.EQ.1 ) THEN
00447 *
00448 *              1-by-1 pivot block D(k): column k now holds
00449 *
00450 *              W(k) = L(k)*D(k)
00451 *
00452 *              where L(k) is the k-th column of L
00453 *
00454                IF( K.LT.N ) THEN
00455 *
00456 *                 Perform a rank-1 update of A(k+1:n,k+1:n) as
00457 *
00458 *                 A := A - L(k)*D(k)*L(k)' = A - W(k)*(1/D(k))*W(k)'
00459 *
00460                   R1 = CONE / A( K, K )
00461                   CALL ZSYR( UPLO, N-K, -R1, A( K+1, K ), 1,
00462      $                       A( K+1, K+1 ), LDA )
00463 *
00464 *                 Store L(k) in column K
00465 *
00466                   CALL ZSCAL( N-K, R1, A( K+1, K ), 1 )
00467                END IF
00468             ELSE
00469 *
00470 *              2-by-2 pivot block D(k)
00471 *
00472                IF( K.LT.N-1 ) THEN
00473 *
00474 *                 Perform a rank-2 update of A(k+2:n,k+2:n) as
00475 *
00476 *                 A := A - ( L(k) L(k+1) )*D(k)*( L(k) L(k+1) )'
00477 *                    = A - ( W(k) W(k+1) )*inv(D(k))*( W(k) W(k+1) )'
00478 *
00479 *                 where L(k) and L(k+1) are the k-th and (k+1)-th
00480 *                 columns of L
00481 *
00482                   D21 = A( K+1, K )
00483                   D11 = A( K+1, K+1 ) / D21
00484                   D22 = A( K, K ) / D21
00485                   T = CONE / ( D11*D22-CONE )
00486                   D21 = T / D21
00487 *
00488                   DO 60 J = K + 2, N
00489                      WK = D21*( D11*A( J, K )-A( J, K+1 ) )
00490                      WKP1 = D21*( D22*A( J, K+1 )-A( J, K ) )
00491                      DO 50 I = J, N
00492                         A( I, J ) = A( I, J ) - A( I, K )*WK -
00493      $                              A( I, K+1 )*WKP1
00494    50                CONTINUE
00495                      A( J, K ) = WK
00496                      A( J, K+1 ) = WKP1
00497    60             CONTINUE
00498                END IF
00499             END IF
00500          END IF
00501 *
00502 *        Store details of the interchanges in IPIV
00503 *
00504          IF( KSTEP.EQ.1 ) THEN
00505             IPIV( K ) = KP
00506          ELSE
00507             IPIV( K ) = -KP
00508             IPIV( K+1 ) = -KP
00509          END IF
00510 *
00511 *        Increase K and return to the start of the main loop
00512 *
00513          K = K + KSTEP
00514          GO TO 40
00515 *
00516       END IF
00517 *
00518    70 CONTINUE
00519       RETURN
00520 *
00521 *     End of ZSYTF2
00522 *
00523       END
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