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    1: *> \brief \b ZHPTRD
    2: *
    3: *  =========== DOCUMENTATION ===========
    4: *
    5: * Online html documentation available at 
    6: *            http://www.netlib.org/lapack/explore-html/ 
    7: *
    8: *> \htmlonly
    9: *> Download ZHPTRD + dependencies 
   10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zhptrd.f"> 
   11: *> [TGZ]</a> 
   12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zhptrd.f"> 
   13: *> [ZIP]</a> 
   14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zhptrd.f"> 
   15: *> [TXT]</a>
   16: *> \endhtmlonly 
   17: *
   18: *  Definition:
   19: *  ===========
   20: *
   21: *       SUBROUTINE ZHPTRD( UPLO, N, AP, D, E, TAU, INFO )
   22: * 
   23: *       .. Scalar Arguments ..
   24: *       CHARACTER          UPLO
   25: *       INTEGER            INFO, N
   26: *       ..
   27: *       .. Array Arguments ..
   28: *       DOUBLE PRECISION   D( * ), E( * )
   29: *       COMPLEX*16         AP( * ), TAU( * )
   30: *       ..
   31: *  
   32: *
   33: *> \par Purpose:
   34: *  =============
   35: *>
   36: *> \verbatim
   37: *>
   38: *> ZHPTRD reduces a complex Hermitian matrix A stored in packed form to
   39: *> real symmetric tridiagonal form T by a unitary similarity
   40: *> transformation: Q**H * A * Q = T.
   41: *> \endverbatim
   42: *
   43: *  Arguments:
   44: *  ==========
   45: *
   46: *> \param[in] UPLO
   47: *> \verbatim
   48: *>          UPLO is CHARACTER*1
   49: *>          = 'U':  Upper triangle of A is stored;
   50: *>          = 'L':  Lower triangle of A is stored.
   51: *> \endverbatim
   52: *>
   53: *> \param[in] N
   54: *> \verbatim
   55: *>          N is INTEGER
   56: *>          The order of the matrix A.  N >= 0.
   57: *> \endverbatim
   58: *>
   59: *> \param[in,out] AP
   60: *> \verbatim
   61: *>          AP is COMPLEX*16 array, dimension (N*(N+1)/2)
   62: *>          On entry, the upper or lower triangle of the Hermitian matrix
   63: *>          A, packed columnwise in a linear array.  The j-th column of A
   64: *>          is stored in the array AP as follows:
   65: *>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
   66: *>          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
   67: *>          On exit, if UPLO = 'U', the diagonal and first superdiagonal
   68: *>          of A are overwritten by the corresponding elements of the
   69: *>          tridiagonal matrix T, and the elements above the first
   70: *>          superdiagonal, with the array TAU, represent the unitary
   71: *>          matrix Q as a product of elementary reflectors; if UPLO
   72: *>          = 'L', the diagonal and first subdiagonal of A are over-
   73: *>          written by the corresponding elements of the tridiagonal
   74: *>          matrix T, and the elements below the first subdiagonal, with
   75: *>          the array TAU, represent the unitary matrix Q as a product
   76: *>          of elementary reflectors. See Further Details.
   77: *> \endverbatim
   78: *>
   79: *> \param[out] D
   80: *> \verbatim
   81: *>          D is DOUBLE PRECISION array, dimension (N)
   82: *>          The diagonal elements of the tridiagonal matrix T:
   83: *>          D(i) = A(i,i).
   84: *> \endverbatim
   85: *>
   86: *> \param[out] E
   87: *> \verbatim
   88: *>          E is DOUBLE PRECISION array, dimension (N-1)
   89: *>          The off-diagonal elements of the tridiagonal matrix T:
   90: *>          E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
   91: *> \endverbatim
   92: *>
   93: *> \param[out] TAU
   94: *> \verbatim
   95: *>          TAU is COMPLEX*16 array, dimension (N-1)
   96: *>          The scalar factors of the elementary reflectors (see Further
   97: *>          Details).
   98: *> \endverbatim
   99: *>
  100: *> \param[out] INFO
  101: *> \verbatim
  102: *>          INFO is INTEGER
  103: *>          = 0:  successful exit
  104: *>          < 0:  if INFO = -i, the i-th argument had an illegal value
  105: *> \endverbatim
  106: *
  107: *  Authors:
  108: *  ========
  109: *
  110: *> \author Univ. of Tennessee 
  111: *> \author Univ. of California Berkeley 
  112: *> \author Univ. of Colorado Denver 
  113: *> \author NAG Ltd. 
  114: *
  115: *> \date November 2011
  116: *
  117: *> \ingroup complex16OTHERcomputational
  118: *
  119: *> \par Further Details:
  120: *  =====================
  121: *>
  122: *> \verbatim
  123: *>
  124: *>  If UPLO = 'U', the matrix Q is represented as a product of elementary
  125: *>  reflectors
  126: *>
  127: *>     Q = H(n-1) . . . H(2) H(1).
  128: *>
  129: *>  Each H(i) has the form
  130: *>
  131: *>     H(i) = I - tau * v * v**H
  132: *>
  133: *>  where tau is a complex scalar, and v is a complex vector with
  134: *>  v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in AP,
  135: *>  overwriting A(1:i-1,i+1), and tau is stored in TAU(i).
  136: *>
  137: *>  If UPLO = 'L', the matrix Q is represented as a product of elementary
  138: *>  reflectors
  139: *>
  140: *>     Q = H(1) H(2) . . . H(n-1).
  141: *>
  142: *>  Each H(i) has the form
  143: *>
  144: *>     H(i) = I - tau * v * v**H
  145: *>
  146: *>  where tau is a complex scalar, and v is a complex vector with
  147: *>  v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in AP,
  148: *>  overwriting A(i+2:n,i), and tau is stored in TAU(i).
  149: *> \endverbatim
  150: *>
  151: *  =====================================================================
  152:       SUBROUTINE ZHPTRD( UPLO, N, AP, D, E, TAU, INFO )
  153: *
  154: *  -- LAPACK computational routine (version 3.4.0) --
  155: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
  156: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
  157: *     November 2011
  158: *
  159: *     .. Scalar Arguments ..
  160:       CHARACTER          UPLO
  161:       INTEGER            INFO, N
  162: *     ..
  163: *     .. Array Arguments ..
  164:       DOUBLE PRECISION   D( * ), E( * )
  165:       COMPLEX*16         AP( * ), TAU( * )
  166: *     ..
  167: *
  168: *  =====================================================================
  169: *
  170: *     .. Parameters ..
  171:       COMPLEX*16         ONE, ZERO, HALF
  172:       PARAMETER          ( ONE = ( 1.0D+0, 0.0D+0 ),
  173:      $                   ZERO = ( 0.0D+0, 0.0D+0 ),
  174:      $                   HALF = ( 0.5D+0, 0.0D+0 ) )
  175: *     ..
  176: *     .. Local Scalars ..
  177:       LOGICAL            UPPER
  178:       INTEGER            I, I1, I1I1, II
  179:       COMPLEX*16         ALPHA, TAUI
  180: *     ..
  181: *     .. External Subroutines ..
  182:       EXTERNAL           XERBLA, ZAXPY, ZHPMV, ZHPR2, ZLARFG
  183: *     ..
  184: *     .. External Functions ..
  185:       LOGICAL            LSAME
  186:       COMPLEX*16         ZDOTC
  187:       EXTERNAL           LSAME, ZDOTC
  188: *     ..
  189: *     .. Intrinsic Functions ..
  190:       INTRINSIC          DBLE
  191: *     ..
  192: *     .. Executable Statements ..
  193: *
  194: *     Test the input parameters
  195: *
  196:       INFO = 0
  197:       UPPER = LSAME( UPLO, 'U' )
  198:       IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
  199:          INFO = -1
  200:       ELSE IF( N.LT.0 ) THEN
  201:          INFO = -2
  202:       END IF
  203:       IF( INFO.NE.0 ) THEN
  204:          CALL XERBLA( 'ZHPTRD', -INFO )
  205:          RETURN
  206:       END IF
  207: *
  208: *     Quick return if possible
  209: *
  210:       IF( N.LE.0 )
  211:      $   RETURN
  212: *
  213:       IF( UPPER ) THEN
  214: *
  215: *        Reduce the upper triangle of A.
  216: *        I1 is the index in AP of A(1,I+1).
  217: *
  218:          I1 = N*( N-1 ) / 2 + 1
  219:          AP( I1+N-1 ) = DBLE( AP( I1+N-1 ) )
  220:          DO 10 I = N - 1, 1, -1
  221: *
  222: *           Generate elementary reflector H(i) = I - tau * v * v**H
  223: *           to annihilate A(1:i-1,i+1)
  224: *
  225:             ALPHA = AP( I1+I-1 )
  226:             CALL ZLARFG( I, ALPHA, AP( I1 ), 1, TAUI )
  227:             E( I ) = ALPHA
  228: *
  229:             IF( TAUI.NE.ZERO ) THEN
  230: *
  231: *              Apply H(i) from both sides to A(1:i,1:i)
  232: *
  233:                AP( I1+I-1 ) = ONE
  234: *
  235: *              Compute  y := tau * A * v  storing y in TAU(1:i)
  236: *
  237:                CALL ZHPMV( UPLO, I, TAUI, AP, AP( I1 ), 1, ZERO, TAU,
  238:      $                     1 )
  239: *
  240: *              Compute  w := y - 1/2 * tau * (y**H *v) * v
  241: *
  242:                ALPHA = -HALF*TAUI*ZDOTC( I, TAU, 1, AP( I1 ), 1 )
  243:                CALL ZAXPY( I, ALPHA, AP( I1 ), 1, TAU, 1 )
  244: *
  245: *              Apply the transformation as a rank-2 update:
  246: *                 A := A - v * w**H - w * v**H
  247: *
  248:                CALL ZHPR2( UPLO, I, -ONE, AP( I1 ), 1, TAU, 1, AP )
  249: *
  250:             END IF
  251:             AP( I1+I-1 ) = E( I )
  252:             D( I+1 ) = AP( I1+I )
  253:             TAU( I ) = TAUI
  254:             I1 = I1 - I
  255:    10    CONTINUE
  256:          D( 1 ) = AP( 1 )
  257:       ELSE
  258: *
  259: *        Reduce the lower triangle of A. II is the index in AP of
  260: *        A(i,i) and I1I1 is the index of A(i+1,i+1).
  261: *
  262:          II = 1
  263:          AP( 1 ) = DBLE( AP( 1 ) )
  264:          DO 20 I = 1, N - 1
  265:             I1I1 = II + N - I + 1
  266: *
  267: *           Generate elementary reflector H(i) = I - tau * v * v**H
  268: *           to annihilate A(i+2:n,i)
  269: *
  270:             ALPHA = AP( II+1 )
  271:             CALL ZLARFG( N-I, ALPHA, AP( II+2 ), 1, TAUI )
  272:             E( I ) = ALPHA
  273: *
  274:             IF( TAUI.NE.ZERO ) THEN
  275: *
  276: *              Apply H(i) from both sides to A(i+1:n,i+1:n)
  277: *
  278:                AP( II+1 ) = ONE
  279: *
  280: *              Compute  y := tau * A * v  storing y in TAU(i:n-1)
  281: *
  282:                CALL ZHPMV( UPLO, N-I, TAUI, AP( I1I1 ), AP( II+1 ), 1,
  283:      $                     ZERO, TAU( I ), 1 )
  284: *
  285: *              Compute  w := y - 1/2 * tau * (y**H *v) * v
  286: *
  287:                ALPHA = -HALF*TAUI*ZDOTC( N-I, TAU( I ), 1, AP( II+1 ),
  288:      $                 1 )
  289:                CALL ZAXPY( N-I, ALPHA, AP( II+1 ), 1, TAU( I ), 1 )
  290: *
  291: *              Apply the transformation as a rank-2 update:
  292: *                 A := A - v * w**H - w * v**H
  293: *
  294:                CALL ZHPR2( UPLO, N-I, -ONE, AP( II+1 ), 1, TAU( I ), 1,
  295:      $                     AP( I1I1 ) )
  296: *
  297:             END IF
  298:             AP( II+1 ) = E( I )
  299:             D( I ) = AP( II )
  300:             TAU( I ) = TAUI
  301:             II = I1I1
  302:    20    CONTINUE
  303:          D( N ) = AP( II )
  304:       END IF
  305: *
  306:       RETURN
  307: *
  308: *     End of ZHPTRD
  309: *
  310:       END