# p?unmbr

Multiplies a general matrix by one of the unitary transformation matrices from a reduction to bidiagonal form determined by p?gebrd.

## Syntax

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

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

## Description

If `vect = 'Q'`, the routine overwrites the general complex 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 = 'C': | QH*sub(C) | sub(C)*QH

If `vect = 'P'`, the routine overwrites sub(C) with

| side ='L' | side ='R'
trans = 'N': | P*sub(C) | sub(C)*P
trans = 'C': | PH*sub(C) | sub(C)*PH

Here Q and PH are the unitary distributed matrices determined by p?gebrd when reducing a complex distributed matrix `A(ia:*, ja:*)` to bidiagonal form: `A(ia:*,ja:*) = Q*B*PH`.

Q and PH are defined as products of elementary reflectors H(i) and G(i) respectively.

Let `nq = m` if `side = 'L'` and `nq = n` if `side = 'R'`. Thus nq is the order of the unitary matrix Q or PH that is applied.

If `vect = 'Q'`, `A(ia:*, ja:*)` is assumed to have been an nq-by-k matrix:

If `nq ≥ k`, `Q = H(1) H(2)... H(k)`;

If `nq < k`, `Q = H(1) H(2)... H(nq-1)`.

If `vect = 'P'`, `A(ia:*, ja:*)` is assumed to have been a k-by-nq matrix:

If `k < nq`, `P = G(1) G(2)... G(k)`;

If `k ≥ nq`, `P = G(1) G(2)... G(nq-1)`.

## Input Parameters

vect

  (global) CHARACTER.

  If `vect ='Q'`, then Q or QH is applied.

  If `vect ='P'`, then P or PH is applied.

side

  (global) CHARACTER.

  If `side ='L'`, then Q or QH, P or PH is applied from the left.

  If `side ='R'`, then Q or QH, P or PH is applied from the right.

trans

  (global) CHARACTER.

  If `trans = 'N'`, no transpose, Q or P is applied.

  If `trans = 'C'`, conjugate transpose, QH or PH is applied.

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`.

k

  (global) INTEGER.

  If `vect = 'Q'`, the number of columns in the original distributed matrix reduced by p?gebrd;

  If `vect = 'P'`, the number of rows in the original distributed matrix reduced by p?gebrd.

  Constraints: k ≥ 0.

a

  (local)

  COMPLEX for psormbr

  DOUBLE COMPLEX for pdormbr.

  Pointer into the local memory to an array of dimension `(lld_a, LOCc(ja+min(nq,k)-1))` if `vect='Q'`, and `(lld_a, LOCc(ja+nq-1))` if `vect = 'P'`.

  `nq = m` if `side = 'L'`, and `nq = n` otherwise.

  The vectors which define the elementary reflectors H(i) and G(i), whose products determine the matrices Q and P, as returned by p?gebrd.

  If `vect = 'Q'`, `lld_a ≥ max(1, LOCr(ia+nq-1))`;

  If `vect = 'P'`, `lld_a ≥ max(1, LOCr(ia+min(nq, k)-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)

  COMPLEX for pcunmbr

  DOUBLE COMPLEX for pzunmbr.

  Array, DIMENSION `LOCc(ja+min(nq, k)-1)`, if `vect = 'Q'`, and `LOCr(ia+min(nq, k)-1)`, if `vect = 'P'`.

  tau(i) must contain the scalar factor of the elementary reflector H(i) or G(i), which determines Q or P, as returned by p?gebrd in its array argument tauq or taup. tau is tied to the distributed matrix A.

c

  (local) COMPLEX for pcunmbr

  DOUBLE COMPLEX for pzunmbr

  Pointer into the local memory to an array of dimension `(lld_a, LOCc (jc+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)

  COMPLEX for pcunmbr

  DOUBLE COMPLEX for pzunmbr.

  Workspace array of dimension lwork.

lwork

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

  If `side = 'L'`

  `nq = m;`

  if`((vect = 'Q'` and `nq ≥ k)` or `(vect is not equal to 'Q'` and `nq > k))`, `iaa= ia`; `jaa= ja`; `mi= m`; `ni= n`; `icc= ic`; `jcc= jc`;

  else

  iaa= ia+1; `jaa= ja`; mi= m-1; `ni= n`; icc= ic+1; `jcc= jc`;

  end if

  else

  If `side = 'R'`, nq = n;

  `if ((vect = 'Q' and nq ≥ k) or (vect is not equal to 'Q' and nq ≥ k))`,

  iaa= ia; `jaa= ja`; `mi= m`; `ni= n`; `icc= ic`; `jcc= jc`;

  else

  iaa= ia; `jaa= ja+1`; `mi= m`; `ni= n-1`; `icc= ic`; `jcc= jc+1`;

  end if

  end if

  If `vect = 'Q'`,

  If `side = 'L'`, `lwork ≥ max((nb_a*(nb_a-1))/2, (nqc0+mpc0)*nb_a) + nb_a*nb_a`

  else if `side = 'R'`,

  `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

  else if vect is not equal to 'Q',

  if `side = 'L'`,

  `lwork ≥ max((mb_a*(mb_a-1))/2, (mpc0 + max(mqa0+numroc(numroc(mi+iroffc, mb_a, 0, 0, NPROW), mb_a, 0, 0, lcmp), nqc0))*mb_a) + mb_a*mb_a`

  else if `side = 'R'`,

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

  end if

  end if

  where `lcmp = lcm/NPROW, 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)`,

  `iacol = indxg2p(jaa, nb_a, MYCOL, csrc_a, NPCOL)`,

  mqa0 = numroc(mi+icoffa, nb_a, MYCOL, iacol, NPCOL),

  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)`,

  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

  On exit, if `vect='Q'`, sub(C) is overwritten by `Q*sub(C),` or `Q'*sub(C),` or `sub(C)*Q'`, or `sub(C)*Q`; if `vect='P'`, sub(C) is overwritten by `P*sub(C),` or `P'*sub(C),` or `sub(C)*P,` or `sub(C)*P'`.

`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: Singular Value Decomposition