# p?ormtr

Multiplies a general matrix by the orthogonal transformation matrix from a reduction to tridiagonal form determined by p?sytrd.

## Syntax

call psormtr( side, uplo, trans, m, n, a, ia, ja, desca, tau, c, ic, jc, descc, work, lwork, info)

call pdormtr( side, uplo, trans, m, n, a, ia, ja, desca, tau, c, ic, jc, descc, work, lwork, info)

## Description

The routine overwrites the general real distributed m-by-n matrix `sub(C) = C(ic:ic+m-1,jc:jc+n-1)` with

| side ='L' | side ='R'
trans = 'N': | Q*sub(C) | sub(C)*Q
trans = 'T': | QT*sub(C) | sub(C)*QT

where Q is a real orthogonal distributed matrix of order nq, with `nq = m` if `side = 'L'` and `nq = n` if `side = 'R'`.

Q is defined as the product of nq elementary reflectors, as returned by p?sytrd.

If `uplo = 'U'`, `Q = H(nq-1)... H(2) H(1)`;

If `uplo = 'L'`, `Q = H(1) H(2)... H(nq-1)`.

## Input Parameters

side

  (global) CHARACTER

  `='L'`: Q or QT is applied from the left.

  `='R'`: Q or QT is applied from the right.

trans

  (global) CHARACTER

  `='N'`, no transpose, Q is applied.

  `='T'`, transpose, QT is applied.

uplo

  (global) CHARACTER.

  `= 'U'`: Upper triangle of `A(ia:*, ja:*)` contains elementary reflectors from p?sytrd;

  `= 'L'`: Lower triangle of `A(ia:*,ja:*)` contains elementary reflectors from p?sytrd

m

  (global) INTEGER. The number of rows in the distributed matrix sub(C) `(m≥0)`.

n

  (global) INTEGER. The number of columns in the distributed matrix sub(C) `(n≥0)`.

a

  (local)

  REAL for psormtr

  DOUBLE PRECISION for pdormtr.

  Pointer into the local memory to an array of dimension `(lld_a, LOCc(ja+m-1))` if `side='L'`, or `(lld_a, LOCc(ja+n-1))` if `side = 'R'`.

  Contains the vectors which define the elementary reflectors, as returned by p?sytrd.

  If `side='L'`, `lld_a ≥ max(1,LOCr(ia+m-1))`;

  If `side ='R'`, lld_a ≥ max(1, LOCr(ia+n-1)).

ia, ja

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

desca

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

tau

  (local)

  REAL for psormtr

  DOUBLE PRECISION for pdormtr.

  Array, DIMENSION of ltau where

  if `side = 'L'` and uplo = 'U', ltau = LOCc(m_a),

  if `side = 'L'` and uplo = 'L', ltau = LOCc(ja+m-2),

  if `side = 'R'` and uplo = 'U', ltau = LOCc(n_a),

  if `side = 'R'` and uplo = 'L', ltau = LOCc(ja+n-2).

  tau(i) must contain the scalar factor of the elementary reflector H(i), as returned by p?sytrd. tau is tied to the distributed matrix A.

c

  (local) REAL for psormtr

  DOUBLE PRECISION for pdormtr.

  Pointer into the local memory to an array of dimension `(lld_a, LOCc (ja+n-1))`. Contains the local pieces of the distributed matrix sub (C).

ic, jc

  (global) INTEGER. The row and column indices in the global array c indicating the first row and the first column of the submatrix C, respectively.

descc

  (global and local) INTEGER array, dimension (dlen_). The array descriptor for the distributed matrix C.

work

  (local)

  REAL for psormtr

  DOUBLE PRECISION for pdormtr.

  Workspace array of dimension lwork.

lwork

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

  `if `uplo = 'U'``,

  `iaa= ia; jaa= ja+1, `icc= ic`; jcc= jc`;

  else `uplo = 'L'`,

  `iaa= ia+1, jaa= ja`;

  If `side = 'L'`,

  `icc= ic+1; jcc= jc`;

  else `icc= ic; jcc= jc+1`;

  end if

  end if

  If `side = 'L'`,

  `mi= m-1; ni= n`

  `lwork ≥ max((nb_a*(nb_a-1))/2, (nqc0 + mpc0)*nb_a) + nb_a*nb_a`

  else

  If `side = 'R'`,

  `mi= m; mi = n-1`;

  `lwork ≥ max((nb_a*(nb_a-1))/2, (nqc0 + max(npa0+numroc(numroc(ni+icoffc, nb_a, 0, 0, NPCOL), nb_a, 0, 0, lcmq), mpc0))*nb_a)+ nb_a*nb_a`

  end if

  where `lcmq = lcm/NPCOL with lcm = ilcm(NPROW, NPCOL)`,

  `iroffa = mod(iaa-1, mb_a)`,

  `icoffa = mod(jaa-1, nb_a)`,

  `iarow = indxg2p(iaa, mb_a, MYROW, rsrc_a, NPROW)`,

  `npa0 = numroc(ni+iroffa, mb_a, MYROW, iarow, NPROW)`,

  `iroffc = mod(icc-1, mb_c)`,

  `icoffc = mod(jcc-1, nb_c)`,

  `icrow = indxg2p(icc, mb_c, MYROW, rsrc_c, NPROW)`,

  `iccol = indxg2p(jcc, nb_c, MYCOL, csrc_c, NPCOL)`,

  `mpc0 = numroc(mi+iroffc, mb_c, MYROW, icrow, NPROW)`,

  `nqc0 = numroc(ni+icoffc, nb_c, MYCOL, iccol, NPCOL)`,

  ilcm, 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

c

  Overwritten by the product Q*sub(C), or Q'*sub(C), or sub(C)*Q', or sub(C)*Q.

`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-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`.

Parent topic: Symmetric Eigenproblems