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chetrd(3P)		    Sun Performance Library		    chetrd(3P)

NAME
       chetrd - reduce a complex Hermitian matrix A to real symmetric tridiag‐
       onal form T by a unitary similarity transformation

SYNOPSIS
       SUBROUTINE CHETRD(UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO)

       CHARACTER * 1 UPLO
       COMPLEX A(LDA,*), TAU(*), WORK(*)
       INTEGER N, LDA, LWORK, INFO
       REAL D(*), E(*)

       SUBROUTINE CHETRD_64(UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO)

       CHARACTER * 1 UPLO
       COMPLEX A(LDA,*), TAU(*), WORK(*)
       INTEGER*8 N, LDA, LWORK, INFO
       REAL D(*), E(*)

   F95 INTERFACE
       SUBROUTINE HETRD(UPLO, [N], A, [LDA], D, E, TAU, [WORK], [LWORK],
	      [INFO])

       CHARACTER(LEN=1) :: UPLO
       COMPLEX, DIMENSION(:) :: TAU, WORK
       COMPLEX, DIMENSION(:,:) :: A
       INTEGER :: N, LDA, LWORK, INFO
       REAL, DIMENSION(:) :: D, E

       SUBROUTINE HETRD_64(UPLO, [N], A, [LDA], D, E, TAU, [WORK], [LWORK],
	      [INFO])

       CHARACTER(LEN=1) :: UPLO
       COMPLEX, DIMENSION(:) :: TAU, WORK
       COMPLEX, DIMENSION(:,:) :: A
       INTEGER(8) :: N, LDA, LWORK, INFO
       REAL, DIMENSION(:) :: D, E

   C INTERFACE
       #include <sunperf.h>

       void chetrd(char uplo, int n, complex *a, int lda, float *d, float  *e,
		 complex *tau, int *info);

       void chetrd_64(char uplo, long n, complex *a, long lda, float *d, float
		 *e, complex *tau, long *info);

PURPOSE
       chetrd reduces a complex Hermitian matrix A to real symmetric tridiago‐
       nal form T by a unitary similarity transformation: Q**H * A * Q = T.

ARGUMENTS
       UPLO (input)
		 = 'U':	 Upper triangle of A is stored;
		 = 'L':	 Lower triangle of A is stored.

       N (input) The order of the matrix A.  N >= 0.

       A (input/output)
		 On entry, the Hermitian matrix A.  If UPLO = 'U', the leading
		 N-by-N upper triangular part of A contains the upper triangu‐
		 lar  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	corre‐
		 sponding  elements  of the tridiagonal matrix T, and the ele‐
		 ments above the first superdiagonal, with the array TAU, rep‐
		 resent	 the  unitary  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 subdi‐
		 agonal, with the array TAU, represent the unitary matrix Q as
		 a product of elementary reflectors. See Further Details.

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

       D (output)
		 The diagonal elements of the tridiagonal  matrix  T:  D(i)  =
		 A(i,i).

       E (output)
		 The off-diagonal elements of the tridiagonal matrix T: E(i) =
		 A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.

       TAU (output)
		 The scalar factors of the elementary reflectors (see  Further
		 Details).

       WORK (workspace)
		 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

       LWORK (input)
		 The  dimension	 of  the array WORK.  LWORK >= 1.  For optimum
		 performance LWORK >= N*NB, where NB is the optimal blocksize.

		 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.

       INFO (output)
		 = 0:  successful exit
		 < 0:  if INFO = -i, the i-th argument had an illegal value

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 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(1:i-1,i+1), 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(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(i+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':

	 (  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).

				  6 Mar 2009			    chetrd(3P)
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