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    1: *> \brief \b DSPTRD
    2: *
    3: *  =========== DOCUMENTATION ===========
    4: *
    5: * Online html documentation available at 
    6: *            http://www.netlib.org/lapack/explore-html/ 
    7: *
    8: *> \htmlonly
    9: *> Download DSPTRD + dependencies 
   10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsptrd.f"> 
   11: *> [TGZ]</a> 
   12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsptrd.f"> 
   13: *> [ZIP]</a> 
   14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsptrd.f"> 
   15: *> [TXT]</a>
   16: *> \endhtmlonly 
   17: *
   18: *  Definition:
   19: *  ===========
   20: *
   21: *       SUBROUTINE DSPTRD( 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   AP( * ), D( * ), E( * ), TAU( * )
   29: *       ..
   30: *  
   31: *
   32: *> \par Purpose:
   33: *  =============
   34: *>
   35: *> \verbatim
   36: *>
   37: *> DSPTRD reduces a real symmetric matrix A stored in packed form to
   38: *> symmetric tridiagonal form T by an orthogonal similarity
   39: *> transformation: Q**T * A * Q = T.
   40: *> \endverbatim
   41: *
   42: *  Arguments:
   43: *  ==========
   44: *
   45: *> \param[in] UPLO
   46: *> \verbatim
   47: *>          UPLO is CHARACTER*1
   48: *>          = 'U':  Upper triangle of A is stored;
   49: *>          = 'L':  Lower triangle of A is stored.
   50: *> \endverbatim
   51: *>
   52: *> \param[in] N
   53: *> \verbatim
   54: *>          N is INTEGER
   55: *>          The order of the matrix A.  N >= 0.
   56: *> \endverbatim
   57: *>
   58: *> \param[in,out] AP
   59: *> \verbatim
   60: *>          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
   61: *>          On entry, the upper or lower triangle of the symmetric matrix
   62: *>          A, packed columnwise in a linear array.  The j-th column of A
   63: *>          is stored in the array AP as follows:
   64: *>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
   65: *>          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
   66: *>          On exit, if UPLO = 'U', the diagonal and first superdiagonal
   67: *>          of A are overwritten by the corresponding elements of the
   68: *>          tridiagonal matrix T, and the elements above the first
   69: *>          superdiagonal, with the array TAU, represent the orthogonal
   70: *>          matrix Q as a product of elementary reflectors; if UPLO
   71: *>          = 'L', the diagonal and first subdiagonal of A are over-
   72: *>          written by the corresponding elements of the tridiagonal
   73: *>          matrix T, and the elements below the first subdiagonal, with
   74: *>          the array TAU, represent the orthogonal matrix Q as a product
   75: *>          of elementary reflectors. See Further Details.
   76: *> \endverbatim
   77: *>
   78: *> \param[out] D
   79: *> \verbatim
   80: *>          D is DOUBLE PRECISION array, dimension (N)
   81: *>          The diagonal elements of the tridiagonal matrix T:
   82: *>          D(i) = A(i,i).
   83: *> \endverbatim
   84: *>
   85: *> \param[out] E
   86: *> \verbatim
   87: *>          E is DOUBLE PRECISION array, dimension (N-1)
   88: *>          The off-diagonal elements of the tridiagonal matrix T:
   89: *>          E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
   90: *> \endverbatim
   91: *>
   92: *> \param[out] TAU
   93: *> \verbatim
   94: *>          TAU is DOUBLE PRECISION array, dimension (N-1)
   95: *>          The scalar factors of the elementary reflectors (see Further
   96: *>          Details).
   97: *> \endverbatim
   98: *>
   99: *> \param[out] INFO
  100: *> \verbatim
  101: *>          INFO is INTEGER
  102: *>          = 0:  successful exit
  103: *>          < 0:  if INFO = -i, the i-th argument had an illegal value
  104: *> \endverbatim
  105: *
  106: *  Authors:
  107: *  ========
  108: *
  109: *> \author Univ. of Tennessee 
  110: *> \author Univ. of California Berkeley 
  111: *> \author Univ. of Colorado Denver 
  112: *> \author NAG Ltd. 
  113: *
  114: *> \date November 2011
  115: *
  116: *> \ingroup doubleOTHERcomputational
  117: *
  118: *> \par Further Details:
  119: *  =====================
  120: *>
  121: *> \verbatim
  122: *>
  123: *>  If UPLO = 'U', the matrix Q is represented as a product of elementary
  124: *>  reflectors
  125: *>
  126: *>     Q = H(n-1) . . . H(2) H(1).
  127: *>
  128: *>  Each H(i) has the form
  129: *>
  130: *>     H(i) = I - tau * v * v**T
  131: *>
  132: *>  where tau is a real scalar, and v is a real vector with
  133: *>  v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in AP,
  134: *>  overwriting A(1:i-1,i+1), and tau is stored in TAU(i).
  135: *>
  136: *>  If UPLO = 'L', the matrix Q is represented as a product of elementary
  137: *>  reflectors
  138: *>
  139: *>     Q = H(1) H(2) . . . H(n-1).
  140: *>
  141: *>  Each H(i) has the form
  142: *>
  143: *>     H(i) = I - tau * v * v**T
  144: *>
  145: *>  where tau is a real scalar, and v is a real vector with
  146: *>  v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in AP,
  147: *>  overwriting A(i+2:n,i), and tau is stored in TAU(i).
  148: *> \endverbatim
  149: *>
  150: *  =====================================================================
  151:       SUBROUTINE DSPTRD( UPLO, N, AP, D, E, TAU, INFO )
  152: *
  153: *  -- LAPACK computational routine (version 3.4.0) --
  154: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
  155: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
  156: *     November 2011
  157: *
  158: *     .. Scalar Arguments ..
  159:       CHARACTER          UPLO
  160:       INTEGER            INFO, N
  161: *     ..
  162: *     .. Array Arguments ..
  163:       DOUBLE PRECISION   AP( * ), D( * ), E( * ), TAU( * )
  164: *     ..
  165: *
  166: *  =====================================================================
  167: *
  168: *     .. Parameters ..
  169:       DOUBLE PRECISION   ONE, ZERO, HALF
  170:       PARAMETER          ( ONE = 1.0D0, ZERO = 0.0D0,
  171:      $                   HALF = 1.0D0 / 2.0D0 )
  172: *     ..
  173: *     .. Local Scalars ..
  174:       LOGICAL            UPPER
  175:       INTEGER            I, I1, I1I1, II
  176:       DOUBLE PRECISION   ALPHA, TAUI
  177: *     ..
  178: *     .. External Subroutines ..
  179:       EXTERNAL           DAXPY, DLARFG, DSPMV, DSPR2, XERBLA
  180: *     ..
  181: *     .. External Functions ..
  182:       LOGICAL            LSAME
  183:       DOUBLE PRECISION   DDOT
  184:       EXTERNAL           LSAME, DDOT
  185: *     ..
  186: *     .. Executable Statements ..
  187: *
  188: *     Test the input parameters
  189: *
  190:       INFO = 0
  191:       UPPER = LSAME( UPLO, 'U' )
  192:       IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
  193:          INFO = -1
  194:       ELSE IF( N.LT.0 ) THEN
  195:          INFO = -2
  196:       END IF
  197:       IF( INFO.NE.0 ) THEN
  198:          CALL XERBLA( 'DSPTRD', -INFO )
  199:          RETURN
  200:       END IF
  201: *
  202: *     Quick return if possible
  203: *
  204:       IF( N.LE.0 )
  205:      $   RETURN
  206: *
  207:       IF( UPPER ) THEN
  208: *
  209: *        Reduce the upper triangle of A.
  210: *        I1 is the index in AP of A(1,I+1).
  211: *
  212:          I1 = N*( N-1 ) / 2 + 1
  213:          DO 10 I = N - 1, 1, -1
  214: *
  215: *           Generate elementary reflector H(i) = I - tau * v * v**T
  216: *           to annihilate A(1:i-1,i+1)
  217: *
  218:             CALL DLARFG( I, AP( I1+I-1 ), AP( I1 ), 1, TAUI )
  219:             E( I ) = AP( I1+I-1 )
  220: *
  221:             IF( TAUI.NE.ZERO ) THEN
  222: *
  223: *              Apply H(i) from both sides to A(1:i,1:i)
  224: *
  225:                AP( I1+I-1 ) = ONE
  226: *
  227: *              Compute  y := tau * A * v  storing y in TAU(1:i)
  228: *
  229:                CALL DSPMV( UPLO, I, TAUI, AP, AP( I1 ), 1, ZERO, TAU,
  230:      $                     1 )
  231: *
  232: *              Compute  w := y - 1/2 * tau * (y**T *v) * v
  233: *
  234:                ALPHA = -HALF*TAUI*DDOT( I, TAU, 1, AP( I1 ), 1 )
  235:                CALL DAXPY( I, ALPHA, AP( I1 ), 1, TAU, 1 )
  236: *
  237: *              Apply the transformation as a rank-2 update:
  238: *                 A := A - v * w**T - w * v**T
  239: *
  240:                CALL DSPR2( UPLO, I, -ONE, AP( I1 ), 1, TAU, 1, AP )
  241: *
  242:                AP( I1+I-1 ) = E( I )
  243:             END IF
  244:             D( I+1 ) = AP( I1+I )
  245:             TAU( I ) = TAUI
  246:             I1 = I1 - I
  247:    10    CONTINUE
  248:          D( 1 ) = AP( 1 )
  249:       ELSE
  250: *
  251: *        Reduce the lower triangle of A. II is the index in AP of
  252: *        A(i,i) and I1I1 is the index of A(i+1,i+1).
  253: *
  254:          II = 1
  255:          DO 20 I = 1, N - 1
  256:             I1I1 = II + N - I + 1
  257: *
  258: *           Generate elementary reflector H(i) = I - tau * v * v**T
  259: *           to annihilate A(i+2:n,i)
  260: *
  261:             CALL DLARFG( N-I, AP( II+1 ), AP( II+2 ), 1, TAUI )
  262:             E( I ) = AP( II+1 )
  263: *
  264:             IF( TAUI.NE.ZERO ) THEN
  265: *
  266: *              Apply H(i) from both sides to A(i+1:n,i+1:n)
  267: *
  268:                AP( II+1 ) = ONE
  269: *
  270: *              Compute  y := tau * A * v  storing y in TAU(i:n-1)
  271: *
  272:                CALL DSPMV( UPLO, N-I, TAUI, AP( I1I1 ), AP( II+1 ), 1,
  273:      $                     ZERO, TAU( I ), 1 )
  274: *
  275: *              Compute  w := y - 1/2 * tau * (y**T *v) * v
  276: *
  277:                ALPHA = -HALF*TAUI*DDOT( N-I, TAU( I ), 1, AP( II+1 ),
  278:      $                 1 )
  279:                CALL DAXPY( N-I, ALPHA, AP( II+1 ), 1, TAU( I ), 1 )
  280: *
  281: *              Apply the transformation as a rank-2 update:
  282: *                 A := A - v * w**T - w * v**T
  283: *
  284:                CALL DSPR2( UPLO, N-I, -ONE, AP( II+1 ), 1, TAU( I ), 1,
  285:      $                     AP( I1I1 ) )
  286: *
  287:                AP( II+1 ) = E( I )
  288:             END IF
  289:             D( I ) = AP( II )
  290:             TAU( I ) = TAUI
  291:             II = I1I1
  292:    20    CONTINUE
  293:          D( N ) = AP( II )
  294:       END IF
  295: *
  296:       RETURN
  297: *
  298: *     End of DSPTRD
  299: *
  300:       END