PCHETTRD(3) ScaLAPACK routine of NEC Numeric Library Collection PCHETTRD(3)
NAME
PCHENTRD - reduce a complex Hermitian matrix sub( A ) to Hermitian
tridiagonal form T by an unitary similarity transformation
SYNOPSIS
SUBROUTINE PCHETTRD(
UPLO, N, A, IA, JA, DESCA, D, E, TAU, WORK, LWORK,
INFO )
CHARACTER UPLO
INTEGER IA, INFO, JA, LWORK, N
INTEGER DESCA( * )
REAL D( * ), E( * )
COMPLEX*8 A( * ), TAU( * ), WORK( * )
PURPOSE
PCHETTRD reduces a complex Hermitian matrix sub( A ) to Hermitian
tridiagonal form T by an orthogonal similarity transformation: Q' *
sub( A ) * Q = T, where sub( A ) = A(IA:IA+N-1,JA:JA+N-1).
Notes
=====
Each global data object is described by an associated description vec-
tor. This vector stores the information required to establish the map-
ping between an object element and its corresponding process and memory
location.
Let A be a generic term for any 2D block cyclicly distributed array.
Such a global array has an associated description vector DESCA. In the
following comments, the character _ should be read as "of the global
array".
NOTATION STORED IN EXPLANATION
--------------- -------------- --------------------------------------
DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case,
DTYPE_A = 1.
CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating
the BLACS process grid A is distribu-
ted over. The context itself is glo-
bal, but the handle (the integer
value) may vary.
M_A (global) DESCA( M_ ) The number of rows in the global
array A.
N_A (global) DESCA( N_ ) The number of columns in the global
array A.
MB_A (global) DESCA( MB_ ) The blocking factor used to distribute
the rows of the array.
NB_A (global) DESCA( NB_ ) The blocking factor used to distribute
the columns of the array.
RSRC_A (global) DESCA( RSRC_ ) The process row over which the first
row of the array A is distributed.
CSRC_A (global) DESCA( CSRC_ ) The process column over which the
first column of the array A is
distributed.
LLD_A (local) DESCA( LLD_ ) The leading dimension of the local
array. LLD_A >= MAX(1,LOCr(M_A)).
Let K be the number of rows or columns of a distributed matrix, and
assume that its process grid has dimension p x q.
LOCr( K ) denotes the number of elements of K that a process would
receive if K were distributed over the p processes of its process col-
umn.
Similarly, LOCc( K ) denotes the number of elements of K that a process
would receive if K were distributed over the q processes of its process
row.
The values of LOCr() and LOCc() may be determined via a call to the
ScaLAPACK tool function, NUMROC:
LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ),
LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper
bound for these quantities may be computed by:
LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A
LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A
ARGUMENTS
UPLO (global input) CHARACTER
Specifies whether the upper or lower triangular part of the
symmetric matrix sub( A ) is stored:
= 'U': Upper triangular
= 'L': Lower triangular
N (global input) INTEGER
The number of rows and columns to be operated on, i.e. the
order of the distributed submatrix sub( A ). N >= 0.
A (local input/local output) COMPLEX*8 pointer into the
local memory to an array of dimension (LLD_A,LOCc(JA+N-1)). On
entry, this array contains the local pieces of the Hermitian
distributed matrix sub( A ). If UPLO = 'U', the leading N-by-N
upper triangular part of sub( A ) contains the upper triangular
part of the matrix, and its strictly lower triangular part is
not referenced. If UPLO = 'L', the leading N-by-N lower trian-
gular part of sub( A ) contains the lower triangular part of
the matrix, and its strictly upper triangular part is not ref-
erenced. On exit, if UPLO = 'U', the diagonal and first super-
diagonal of sub( A ) are over- written 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 sub( A ) are
overwritten 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.
IA (global input) INTEGER
The row index in the global array A indicating the first row of
sub( A ).
JA (global input) INTEGER
The column index in the global array A indicating the first
column of sub( A ).
DESCA (global and local input) INTEGER array of dimension DLEN_.
The array descriptor for the distributed matrix A.
D (local output) REAL array, dimension LOCc(JA+N-1)
The diagonal elements of the tridiagonal matrix T: D(i) =
A(i,i). D is tied to the distributed matrix A.
E (local output) REAL array, dimension LOCc(JA+N-1)
if UPLO = 'U', LOCc(JA+N-2) otherwise. The off-diagonal ele-
ments of the tridiagonal matrix T: E(i) = A(i,i+1) if UPLO =
'U', E(i) = A(i+1,i) if UPLO = 'L'. E is tied to the dis-
tributed matrix A.
TAU (local output) COMPLEX*8 array, dimension
LOCc(JA+N-1). This array contains the scalar factors TAU of the
elementary reflectors. TAU is tied to the distributed matrix A.
WORK (local workspace/local output) COMPLEX*8 array, dimension
(LWORK)
On exit, WORK( 1 ) returns the minimal and optimal LWORK.
LWORK (local or global input) INTEGER
The dimension of the array WORK. LWORK >= 2*( ANB+1 )*(
4*NPS+2 ) + NPS
Where:
NPS = MAX( NUMROC( N, 1, 0, 0, NPROW ), 2*ANB )
ANB = PJLAENV( DESCA( CTXT_ ), 3, 'PCHETTRD', 'L', 0, 0,
0, 0 )
NUMROC is a ScaLAPACK tool function;
PJLAENV is a ScaLAPACK envionmental inquiry function
MYROW, MYCOL, NPROW and NPCOL can be determined by calling
the subroutine BLACS_GRIDINFO.
INFO (global output) INTEGER
= 0: successful exit
< 0: If the i-th argument is an array and the j-entry had an
illegal value, then INFO = -(i*100+j), if the i-th argument is
a scalar and had an illegal value, then INFO = -i.
FURTHER DETAILS
If UPLO = 'U', the matrix Q is represented as a product of elementary
reflectors
Q = H(n-1) . . . H(2) H(1).
Each H(i) has the form
H(i) = I - tau * v * v'
where tau is a real scalar, and v is a real vector with
v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
A(ia:ia+i-2,ja+i), and tau in TAU(ja+i-1).
If UPLO = 'L', the matrix Q is represented as a product of elementary
reflectors
Q = H(1) H(2) . . . H(n-1).
Each H(i) has the form
H(i) = I - tau * v * v'
where tau is a real scalar, and v is a real vector with
v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in
A(ia+i+1:ia+n-1,ja+i-1), and tau in TAU(ja+i-1).
The contents of sub( A ) on exit are illustrated by the following exam-
ples with n = 5:
if UPLO = 'U': if UPLO = 'L':
( d e v2 v3 v4 ) ( d )
( d e v3 v4 ) ( e d )
( d e v4 ) ( v1 e d )
( d e ) ( v1 v2 e d )
( d ) ( v1 v2 v3 e d )
where d and e denote diagonal and off-diagonal elements of T, and vi
denotes an element of the vector defining H(i).
ALIGNMENT REQUIREMENTS
The distributed submatrix sub( A ) must verify some alignment proper-
ties, namely the following expression should be true:
( MB_A.EQ.NB_A .AND. IROFFA.EQ.ICOFFA .AND. IROFFA.EQ.0 ) with IROFFA =
MOD( IA-1, MB_A ) and ICOFFA = MOD( JA-1, NB_A ).
ScaLAPACK routine 31 October 2017 PCHETTRD(3)