?gebd2

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/31/2023

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?gebd2

Reduces a general matrix to bidiagonal form using an unblocked algorithm.

Syntax

call sgebd2( m, n, a, lda, d, e, tauq, taup, work, info )

call dgebd2( m, n, a, lda, d, e, tauq, taup, work, info )

call cgebd2( m, n, a, lda, d, e, tauq, taup, work, info )

call zgebd2( m, n, a, lda, d, e, tauq, taup, work, info )

Include Files

mkl.fi

Description

The routine reduces a general m-by-n matrix A to upper or lower bidiagonal form B by an orthogonal (unitary) transformation: Q^T*A*P = B (for real flavors) or Q^H*A*P = B (for complex flavors).

If m≥n, B is upper bidiagonal; if m < n, B is lower bidiagonal.

The routine does not form the matrices Q and P explicitly, but represents them as products of elementary reflectors. if m≥n,

Q = H(1)*H(2)*...*H(n), and P = G(1)*G(2)*...*G(n-1)

if m < n,

Q = H(1)*H(2)*...*H(m-1), and P = G(1)*G(2)*...*G(m)

Each H(i) and G(i) has the form

H(i) = I - tauq*v*v^T and G(i) = I - taup*u*u^T for real flavors, or

H(i) = I - tauq*v*v^H and G(i) = I - taup*u*u^H for complex flavors

where tauq and taup are scalars (real for sgebd2/dgebd2, complex for cgebd2/zgebd2), and v and u are vectors (real for sgebd2/dgebd2, complex for cgebd2/zgebd2).

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 sgebd2

DOUBLE PRECISION for dgebd2

COMPLEX for cgebd2

DOUBLE COMPLEX for zgebd2.

Arrays:

a(lda,*) contains the m-by-n general matrix A to be reduced. The second dimension of a must be at least max(1, n).

work(*) is a workspace array, the dimension of work must be at least max(1, m, n).

lda

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

Output Parameters

a

if m≥n, the diagonal and first super-diagonal of a are overwritten with the upper bidiagonal matrix B. Elements below the diagonal, with the array tauq, represent the orthogonal/unitary matrix Q as a product of elementary reflectors, and elements above the first superdiagonal, with the array taup, represent the orthogonal/unitary matrix p as a product of elementary reflectors.

if m < n, the diagonal and first sub-diagonal of a are overwritten by the lower bidiagonal matrix B. Elements below the first subdiagonal, with the array tauq, represent the orthogonal/unitary matrix Q as a product of elementary reflectors, and elements above the diagonal, with the array taup, represent the orthogonal/unitary matrix p as a product of elementary reflectors.

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 the bidiagonal matrix B: d(i) = a(i, i).

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 the bidiagonal matrix B:

if m≥n, e(i) = a(i, i+1) for i = 1,2,..., n-1;

if m < n, e(i) = a(i+1, i) for i = 1,2,..., m-1.

tauq, taup

REAL for sgebd2

DOUBLE PRECISION for dgebd2

COMPLEX for cgebd2

DOUBLE COMPLEX for zgebd2.

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

Contain scalar factors of the elementary reflectors which represent orthogonal/unitary matrices Q and p, respectively.

info

INTEGER.

If info = 0, the execution is successful.

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

Parent topic: LAPACK Auxiliary Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

?gebd2