# ?gebrd

Reduces a general matrix to bidiagonal form.

## Syntax

FORTRAN 77:

call sgebrd(m, n, a, lda, d, e, tauq, taup, work, lwork, info)

call dgebrd(m, n, a, lda, d, e, tauq, taup, work, lwork, info)

call cgebrd(m, n, a, lda, d, e, tauq, taup, work, lwork, info)

call zgebrd(m, n, a, lda, d, e, tauq, taup, work, lwork, info)

Fortran 95:

call gebrd(a [, d] [,e] [,tauq] [,taup] [,info])

## Description

The routine reduces a general m-by-n matrix A to a bidiagonal matrix B by an orthogonal (unitary) transformation.

If `m ≥ n`, the reduction is given by

where B1 is an n-by-n upper diagonal matrix, Q and P are orthogonal or, for a complex A, unitary matrices; Q1 consists of the first n columns of Q.

If `m < n`, the reduction is given by

`A = Q*B*PH = Q*(B10)*PH = Q1*B1*P1H`,

where B1 is an m-by-m lower diagonal matrix, Q and P are orthogonal or, for a complex A, unitary matrices; P1 consists of the first m rows of P.

The routine does not form the matrices Q and P explicitly, but represents them as products of elementary reflectors. Routines are provided to work with the matrices Q and P in this representation:

If the matrix A is real,

- to compute Q and P explicitly, call ?orgbr.

- to multiply a general matrix by Q or P, call ?ormbr.

If the matrix A is complex,

- to compute Q and P explicitly, call ?ungbr.

- to multiply a general matrix by Q or P, call ?unmbr.

## Input Parameters

m

  INTEGER. The number of rows in the matrix A (`m ≥ 0`).

n

  INTEGER. The number of columns in A (`n ≥ 0`).

a, work

  REAL for sgebrd

  DOUBLE PRECISION for dgebrd

  COMPLEX for cgebrd

  DOUBLE COMPLEX for zgebrd.

  Arrays:

  a(lda,*) contains the matrix A.

  The second dimension of a must be at least `max(1, n)`.

  work is a workspace array, its dimension `max(1, lwork)`.

lda

  INTEGER. The first dimension of a; at least `max(1, m)`.

lwork

  INTEGER.

  The dimension of work; at least `max(1, m, n)`.

  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.

  See Application Notes for the suggested value of lwork.

## Output Parameters

a

  If `m ≥ n`, the diagonal and first super-diagonal of a are overwritten by the upper bidiagonal matrix B. Elements below the diagonal are overwritten by details of Q, and the remaining elements are overwritten by details of P.

  If `m < n`, the diagonal and first sub-diagonal of a are overwritten by the lower bidiagonal matrix B. Elements above the diagonal are overwritten by details of P, and the remaining elements are overwritten by details of Q.

d

  REAL for single-precision flavors

  DOUBLE PRECISION for double-precision flavors.

  Array, DIMENSION at least `max(1, min(m, n))`.

  Contains the diagonal elements of B.

e

  REAL for single-precision flavors

  DOUBLE PRECISION for double-precision flavors.

  Array, DIMENSION at least `max(1, min(m, n) - 1)`. Contains the off-diagonal elements of B.

tauq, taup

  REAL for sgebrd

  DOUBLE PRECISION for dgebrd

  COMPLEX for cgebrd

  DOUBLE COMPLEX for zgebrd.

  Arrays, DIMENSION at least `max (1, min(m, n))`. Contain further details of the matrices Q and P.

`work(1)`

  If `info = 0`, on exit `work(1)` contains the minimum value of lwork required for optimum performance. Use this lwork for subsequent runs.

info

  INTEGER.

  If `info = 0`, the execution is successful.

  If `info = -i`, the i-th parameter had an illegal value.

## Fortran 95 Interface Notes

Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or restorable arguments, see Fortran 95 Interface Conventions.

Specific details for the routine gebrd interface are the following:

a

  Holds the matrix A of size (m,n).

d

  Holds the vector of length min(m,n).

e

  Holds the vector of length min(m,n)-1.

tauq

  Holds the vector of length min(m,n).

taup

  Holds the vector of length min(m,n).

## Application Notes

For better performance, try using `lwork = (m + n)*blocksize`, where blocksize is a machine-dependent value (typically, 16 to 64) required for optimum performance of the blocked algorithm.

If you are in doubt how much workspace to supply, use a generous value of lwork for the first run or set `lwork = -1`.

If you choose the first option and set any of admissible lwork sizes, which is no less than the minimal value described, the routine completes the task, though probably not so fast as with a recommended workspace, and provides the recommended workspace in the first element of the corresponding array work on exit. Use this value (`work(1)`) for subsequent runs.

If you set `lwork = -1`, the routine returns immediately and provides the recommended workspace in the first element of the corresponding array (work). This operation is called a workspace query.

Note that if you set lwork to less than the minimal required value and not -1, the routine returns immediately with an error exit and does not provide any information on the recommended workspace.

The computed matrices Q, B, and P satisfy `QBPH = A + E`, where `||E||2 = c(n)ε ||A||2, c(n)` is a modestly increasing function of n, and ε is the machine precision.

The approximate number of floating-point operations for real flavors is

`(4/3)*n2*(3*m - n) for `m ≥ n`,`

`(4/3)*m2*(3*n - m) for m < n.`

The number of operations for complex flavors is four times greater.

If n is much less than m, it can be more efficient to first form the QR factorization of A by calling ?geqrf and then reduce the factor R to bidiagonal form. This requires approximately `2*n2*(m + n)` floating-point operations.

If m is much less than n, it can be more efficient to first form the LQ factorization of A by calling ?gelqf and then reduce the factor L to bidiagonal form. This requires approximately `2*m2*(m + n)` floating-point operations.

Parent topic: Singular Value Decomposition