# p?hetrd

Reduces a Hermitian matrix to Hermitian tridiagonal form by a unitary similarity transformation.

## Syntax

call pchetrd(uplo, n, a, ia, ja, desca, d, e, tau, work, lwork, info)

call pzhetrd(uplo, n, a, ia, ja, desca, d, e, tau, work, lwork, info)

## Description

The p?hetrd routine reduces a complex Hermitian matrix sub(A) to Hermitian tridiagonal form T by a unitary similarity transformation:

Q'*sub(A)*Q = T

where sub(A) = A(ia:ia+n-1,ja:ja+n-1).

## Input Parameters

uplo

(global) CHARACTER.

Specifies whether the upper or lower triangular part of the Hermitian matrix sub(A) is stored:

If `uplo = 'U'`, upper triangular

If `uplo = 'L'`, lower triangular

n

(global) INTEGER. The order of the distributed matrix sub(A) `(n≥0)`.

a

(local)

COMPLEX for pchetrd

DOUBLE COMPLEX for pzhetrd.

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).

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 triangular part of sub(A) contains the lower triangular part of the matrix, and its strictly upper triangular part is not referenced. (see Application Notes below).

ia, ja

(global) INTEGER. The row and column indices in the global matrix A indicating the first row and the first column of the submatrix A, respectively.

desca

(global and local) INTEGER array of size dlen_. The array descriptor for the distributed matrix A.

work

(local)

COMPLEX for pchetrd

DOUBLE COMPLEX for pzhetrd.

Workspace array of size lwork.

lwork

(local or global) INTEGER, size of work, must be at least:

`lwork≥max(NB*(np +1), 3*NB)`

where `NB = mb_a = nb_a`,

`np = numroc(n, NB, MYROW, iarow, NPROW)`,

`iarow = indxg2p(ia, NB, MYROW, rsrc_a, NPROW)`.

indxg2p and numroc are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine blacs_gridinfo.

If `lwork = -1`, then lwork is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by pxerbla.

## Output Parameters

a

On exit,

If `uplo = 'U'`, the diagonal and first superdiagonal of sub(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 unitary 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 unitary matrix Q as a product of elementary reflectors (see Application Notes below).

d

(local)

REAL for pchetrd

DOUBLE PRECISION for pzhetrd.

Arrays of size `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)

REAL for pchetrd

DOUBLE PRECISION for pzhetrd.

Arrays of size `LOCc(ja+n-1)` if `uplo = 'U'`; `LOCc(ja+n-2)` - otherwise.

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'`.

e is tied to the distributed matrix A.

tau

(local) COMPLEX for pchetrd

DOUBLE COMPLEX for pzhetrd.

Array of size `LOCc(ja+n-1)`. This array contains the scalar factors of the elementary reflectors. tau is tied to the distributed matrix A.

`work(1)`

On exit `work(1)` contains the minimum value of lwork required for optimum performance.

info

(global) INTEGER.

`= 0`: the execution is successful.

`< 0`: if the i-th argument is an array and the j-th 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.

## Application Notes

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(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 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).