DSYTRD_SY2SB reduces a real symmetric matrix A to real symmetric


 band-diagonal form AB by a orthogonal similarity transformation:


 Q**T * A * Q = AB.

UPLO

          UPLO is CHARACTER*1


          = 'U':  Upper triangle of A is stored;


          = 'L':  Lower triangle of A is stored.

N

          N is INTEGER


          The order of the matrix A.  N >= 0.

KD

          KD is INTEGER


          The number of superdiagonals of the reduced matrix if UPLO = 'U',


          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.


          The reduced matrix is stored in the array AB.

A

          A is DOUBLE PRECISION array, dimension (LDA,N)


          On entry, the symmetric matrix A.  If UPLO = 'U', the leading


          N-by-N upper triangular part of A contains the upper


          triangular part of the matrix A, and the strictly lower


          triangular part of A is not referenced.  If UPLO = 'L', the


          leading N-by-N lower triangular part of A contains the lower


          triangular part of the matrix A, and the strictly upper


          triangular part of A is not referenced.


          On exit, if UPLO = 'U', the diagonal and first superdiagonal


          of A are overwritten by the corresponding elements of the


          tridiagonal matrix T, and the elements above the first


          superdiagonal, with the array TAU, represent the orthogonal


          matrix Q as a product of elementary reflectors; if UPLO


          = 'L', the diagonal and first subdiagonal of A are over-


          written by the corresponding elements of the tridiagonal


          matrix T, and the elements below the first subdiagonal, with


          the array TAU, represent the orthogonal matrix Q as a product


          of elementary reflectors. See Further Details.

LDA

          LDA is INTEGER


          The leading dimension of the array A.  LDA >= max(1,N).

AB

          AB is DOUBLE PRECISION array, dimension (LDAB,N)


          On exit, the upper or lower triangle of the symmetric band


          matrix A, stored in the first KD+1 rows of the array.  The


          j-th column of A is stored in the j-th column of the array AB


          as follows:


          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;


          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

LDAB

          LDAB is INTEGER


          The leading dimension of the array AB.  LDAB >= KD+1.

TAU

          TAU is DOUBLE PRECISION array, dimension (N-KD)


          The scalar factors of the elementary reflectors (see Further


          Details).

WORK

          WORK is DOUBLE PRECISION array, dimension (LWORK)


          On exit, if INFO = 0, or if LWORK=-1, 


          WORK(1) returns the size of LWORK.

LWORK

          LWORK is INTEGER


          The dimension of the array WORK which should be calculated


          by a workspace query. LWORK = MAX(1, LWORK_QUERY)


          If LWORK = -1, then a workspace query is assumed; the routine


          only calculates the optimal size of the WORK array, returns


          this value as the first entry of the WORK array, and no error


          message related to LWORK is issued by XERBLA.


          LWORK_QUERY = N*KD + N*max(KD,FACTOPTNB) + 2*KD*KD


          where FACTOPTNB is the blocking used by the QR or LQ


          algorithm, usually FACTOPTNB=128 is a good choice otherwise


          putting LWORK=-1 will provide the size of WORK.

INFO

          INFO is INTEGER


          = 0:  successful exit


          < 0:  if INFO = -i, the i-th argument had an illegal value

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

November 2017

  Implemented by Azzam Haidar.


  All details are available on technical report, SC11, SC13 papers.


  Azzam Haidar, Hatem Ltaief, and Jack Dongarra.


  Parallel reduction to condensed forms for symmetric eigenvalue problems


  using aggregated fine-grained and memory-aware kernels. In Proceedings


  of 2011 International Conference for High Performance Computing,


  Networking, Storage and Analysis (SC '11), New York, NY, USA,


  Article 8 , 11 pages.


  http://doi.acm.org/10.1145/2063384.2063394


  A. Haidar, J. Kurzak, P. Luszczek, 2013.


  An improved parallel singular value algorithm and its implementation 


  for multicore hardware, In Proceedings of 2013 International Conference


  for High Performance Computing, Networking, Storage and Analysis (SC '13).


  Denver, Colorado, USA, 2013.


  Article 90, 12 pages.


  http://doi.acm.org/10.1145/2503210.2503292


  A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra.


  A novel hybrid CPU-GPU generalized eigensolver for electronic structure 


  calculations based on fine-grained memory aware tasks.


  International Journal of High Performance Computing Applications.


  Volume 28 Issue 2, Pages 196-209, May 2014.


  http://hpc.sagepub.com/content/28/2/196 

  If UPLO = 'U', the matrix Q is represented as a product of elementary


  reflectors


     Q = H(k)**T . . . H(2)**T H(1)**T, where k = n-kd.


  Each H(i) has the form


     H(i) = I - tau * v * v**T


  where tau is a real scalar, and v is a real vector with


  v(1:i+kd-1) = 0 and v(i+kd) = 1; conjg(v(i+kd+1:n)) is stored on exit in


  A(i,i+kd+1:n), and tau in TAU(i).


  If UPLO = 'L', the matrix Q is represented as a product of elementary


  reflectors


     Q = H(1) H(2) . . . H(k), where k = n-kd.


  Each H(i) has the form


     H(i) = I - tau * v * v**T


  where tau is a real scalar, and v is a real vector with


  v(kd+1:i) = 0 and v(i+kd+1) = 1; v(i+kd+2:n) is stored on exit in


  A(i+kd+2:n,i), and tau in TAU(i).


  The contents of A on exit are illustrated by the following examples


  with n = 5:


  if UPLO = 'U':                       if UPLO = 'L':


    (  ab  ab/v1  v1      v1     v1    )              (  ab                            )


    (      ab     ab/v2   v2     v2    )              (  ab/v1  ab                     )


    (             ab      ab/v3  v3    )              (  v1     ab/v2  ab              )


    (                     ab     ab/v4 )              (  v1     v2     ab/v3  ab       )


    (                            ab    )              (  v1     v2     v3     ab/v4 ab )


  where d and e denote diagonal and off-diagonal elements of T, and vi


  denotes an element of the vector defining H(i)..fi

Definition at line 245 of file dsytrd_sy2sb.f.