p?hentrd
p?hentrd
Reduces a complex Hermitian matrix to Hermitian tridiagonal form.
Syntax
void pchentrd
(
const
char*
uplo
,
const
MKL_INT*
n
,
MKL_Complex8*
a
,
const
MKL_INT*
ia
,
const
MKL_INT*
ja
,
const
MKL_INT*
desca
,
float*
d
,
float*
e
,
MKL_Complex8*
tau
,
MKL_Complex8*
work
,
const
MKL_INT*
lwork
,
float*
rwork
,
const
MKL_INT*
lrwork
,
MKL_INT*
info
);
void pzhentrd
(
const
char*
uplo
,
const
MKL_INT*
n
,
MKL_Complex16*
a
,
const
MKL_INT*
ia
,
const
MKL_INT*
ja
,
const
MKL_INT*
desca
,
double*
d
,
double*
e
,
MKL_Complex16*
tau
,
MKL_Complex16*
work
,
const
MKL_INT*
lwork
,
double*
rwork
,
const
MKL_INT*
lrwork
,
MKL_INT*
info
);
Include Files
- mkl_scalapack.h
Description
p?hentrd
is a prototype version of p?hetrd
which uses tailored codes (either the serial, ?hetrd
, or the parallel code, p?hettrd
) when adequate workspace is provided.p?hentrd
reduces a complex Hermitian matrix sub( A
) to Hermitian tridiagonal form T
by an unitary similarity transformation: Q
' * sub( A
) * Q
= T
, where sub( A
) = A
(ia
:ia
+n
-1,ja
:ja
+n
-1).p?hentrd
is faster than p?hetrd
on almost all matrices, particularly small ones (i.e. n
< 500 * sqrt(P) ), provided that enough workspace is available to use the tailored codes.The tailored codes provide performance that is essentially independent of the input data layout.
The tailored codes place no restrictions on
ia
, ja
, MB or NB. At present, ia
, ja
, MB and NB are restricted to those values allowed by p?hetrd
to keep the interface simple (see the Application Notes section for more information about the restrictions).Input Parameters
- uplo
- (global)Specifies whether the upper or lower triangular part of the Hermitian matrix sub(A) is stored:= 'U': Upper triangular= 'L': Lower triangular
- n
- (global)The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub(A).n>= 0.
- a
- (local)Pointer into the local memory to an array of size.lld_a*LOCc(ja+n-1)On entry, this array contains the local pieces of the Hermitian distributed matrix sub(A). Ifuplo= 'U', the leadingn-by-nupper triangular part of sub(A) contains the upper triangular part of the matrix, and its strictly lower triangular part is not referenced. Ifuplo= 'L', the leadingn-by-nlower triangular part of sub(A) contains the lower triangular part of the matrix, and its strictly upper triangular part is not referenced.
- ia
- (global)The row index in the global arrayaindicating the first row of sub(A).
- ja
- (global)The column index in the global arrayaindicating the first column of sub(A).
- desca
- (global and local)Array of sizedlen_.The array descriptor for the distributed matrixA.
- work
- (local)Array, size (lwork)
- lwork
- (local or global)The size of the arraywork.lworkis local input and must be at leastlwork>= MAX( NB * ( NP +1 ), 3 * NB ).For optimal performance, greater workspace is needed:lwork>= 2*(ANB+1 )*( 4*NPS+2 ) + (NPS+ 4 ) *NPSANB=pjlaenv(ICTXT, 3, 'p?hettrd', 'L', 0, 0, 0, 0 )ICTXT=desca(ctxt_)SQNPC= INT(sqrt( REAL(NPROW*NPCOL) ) )NPS= MAX(numroc(n, 1, 0, 0,SQNPC), 2*ANB)numrocis a ScaLAPACK tool function.pjlaenvis a ScaLAPACK environmental inquiry function.NPROWandNPCOLcan be determined by calling the subroutineblacs_gridinfo.
- rwork
- (local)Array, size (lrwork)
- lrwork
- (local or global)The size of the arrayrwork.lrworkis local input and must be at leastlrwork>= 1.For optimal performance, greater workspace is needed, i.e.lrwork>= MAX( 2 *n)
Output Parameters
- a
- On exit, ifuplo= 'U', the diagonal and first superdiagonal of sub(A) are overwritten by the corresponding elements of the tridiagonal matrixT, and the elements above the first superdiagonal, with the arraytau, represent the unitary matrixQas a product of elementary reflectors; ifuplo= 'L', the diagonal and first subdiagonal of sub(A) are overwritten by the corresponding elements of the tridiagonal matrixT, and the elements below the first subdiagonal, with the arraytau, represent the unitary matrixQas a product of elementary reflectors. See Application Notes.
- d
- (local)Array, size LOCc(ja+n-1)The diagonal elements of the tridiagonal matrixT:d[=i- 1]A(i,i).dis tied to the distributed matrixA.
- e
- (local)Array, size LOCc(ja+n-1) ifuplo= 'U', LOCc(ja+n-2) otherwise.The off-diagonal elements of the tridiagonal matrixT:e[=i- 1]A(i,i+1) ifuplo= 'U',e[=i- 1]A(i+1,i) ifuplo= 'L'.eis tied to the distributed matrixA.
- tau
- (local)Array, size LOCc(ja+n-1).This array contains the scalar factorstauof the elementary reflectors.tauis tied to the distributed matrixA.
- work
- On exit,work[0]returns the optimallwork.
- rwork
- On exit,rwork[0]returns the optimallrwork.
- info
- (global)= 0: successful exit< 0: If thei-th argument is an array and thej-th entry had an illegal value, theninfo= -(i*100+j), if thei-th argument is a scalar and had an illegal value, theninfo= -i.
Application Notes
If
uplo
= 'U', the matrix Q
is represented as a product of elementary reflectorsQ
= H(n
-1) . . . H(2) H(1).Each H(i) has the form
H(i) = I -
tau
* v * v', where tau
is a complex scalar, and v is a complex 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 reflectorsQ
= H(1) H(2) . . . H(n
-1).Each H(i) has the form
H(i) = I -
tau
* v * v', where tau
is a complex scalar, and v is a complex 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 examples with n
= 5:if
uplo
= 'U': 
if
uplo
= 'L': 
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 properties, namely the following expression should be true:(
mb_a
= nb_a
and IROFFA
= ICOFFA
and IROFFA
= 0 ) with IROFFA
= mod( ia
-1, mb_a
), and ICOFFA
= mod( ja
-1, nb_a
).