?orbdb1/?unbdb1

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/31/2023

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?orbdb1/?unbdb1

Simultaneously bidiagonalizes the blocks of a tall and skinny matrix with orthonormal columns.

Syntax

call sorbdb1( m, p, q, x11, ldx11, x21, ldx21, theta, phi, taup1, taup2, tauq1, work, lwork, info )

call dorbdb1( m, p, q, x11, ldx11, x21, ldx21, theta, phi, taup1, taup2, tauq1, work, lwork, info )

call cunbdb1( m, p, q, x11, ldx11, x21, ldx21, theta, phi, taup1, taup2, tauq1, work, lwork, info )

call zunbdb1( m, p, q, x11, ldx11, x21, ldx21, theta, phi, taup1, taup2, tauq1, work, lwork, info )

Include Files

mkl.fi, lapack.f90

Description

The routines ?orbdb1/?unbdb1 simultaneously bidiagonalize the blocks of a tall and skinny matrix X with orthonormal columns:

The size of x₁₁ is p by q, and x₁₂ is (m - p) by q. q must not be larger than p, m-p, or m-q.

Tall and Skinny Matrix Routines
`q`≤ min(`p`, `m` - `p`, `m` - `q`)	?orbdb1/?unbdb1
`p`≤ min(`q`, `m` - `p`, `m` - `q`)	?orbdb2/?unbdb2
`m` - `p`≤ min(`p`, `q`, `m` - `q`)	?orbdb3/?unbdb3
`m` - `q`≤ min(`p`, `q`, `m` - `p`)	?orbdb4/?unbdb4

The orthogonal/unitary matrices p₁, p₂, and q₁ are p-by-p, (m-p)-by-(m-p), (m-q)-by-(m-q), respectively.

p₁, p₂, and q₁ are represented as products of elementary reflectors. See the description of ?orcsd2by1/?uncsd2by1 for details on generating p₁, p₂, and q₁ using ?orgqr and ?orglq.

The upper-bidiagonal matrices b₁₁ and b₁₂ of size q by q are represented implicitly by angles theta(1), ..., theta(q) and phi(1), ..., phi(q-1). Every entry in each bidiagonal band is a product of a sine or cosine of theta with a sine or cosine of phi. See [Sutton09] or the description of ?orcsd/?uncsd for details.

Input Parameters

m

INTEGER. The number of rows in x₁₁ plus the number of rows in x₂₁.

p

INTEGER. The number of rows in x₁₁. 0 ≤p≤m.

q

INTEGER. The number of columns in x₁₁ and x₂₁. 0 ≤q≤ min(p,m-p,m-q).

x11

REAL for sorbdb1

DOUBLE PRECISION for dorbdb1

COMPLEX for cunbdb1

DOUBLE COMPLEX for zunbdb1

Array, DIMENSION (ldx11,q).

On entry, the top block of the orthogonal/unitary matrix to be reduced.

ldx11

INTEGER. The leading dimension of the array X₁₁. ldx11≥p.

x21

REAL for sorbdb1

DOUBLE PRECISION for dorbdb1

COMPLEX for cunbdb1

DOUBLE COMPLEX for zunbdb1

Array, DIMENSION (ldx21,q).

On entry, the bottom block of the orthogonal/unitary matrix to be reduced.

ldx21

INTEGER. The leading dimension of the array X₂₁. ldx21≥m-p.

work

REAL for sorbdb1

DOUBLE PRECISION for dorbdb1

COMPLEX for cunbdb1

DOUBLE COMPLEX for zunbdb1

Workspace array, DIMENSION (lwork).

lwork

INTEGER. The size of the work array. lwork≥m-q

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.

Output Parameters

x11

The columns of tril(x11) specify reflectors for p₁ and the rows of triu(x11,1) specify reflectors for q₁, where tril(A) denotes the lower triangle of A, and triu(A) denotes the upper triangle of A.

x21

On exit, the columns of tril(x21) specify the reflectors for p₂

theta

REAL for sorbdb1

DOUBLE PRECISION for dorbdb1

COMPLEX for cunbdb1

DOUBLE COMPLEX for zunbdb1

Array, DIMENSION (q). The entries of bidiagonal blocks b₁₁ and b₂₁ can be computed from the angles theta and phi. See the Description section for details.

phi

REAL for sorbdb1

DOUBLE PRECISION for dorbdb1

COMPLEX for cunbdb1

DOUBLE COMPLEX for zunbdb1

Array, DIMENSION (q-1). The entries of bidiagonal blocks b₁₁ and b₂₁ can be computed from the angles theta and phi. See the Description section for details.

taup1

REAL for sorbdb1

DOUBLE PRECISION for dorbdb1

COMPLEX for cunbdb1

DOUBLE COMPLEX for zunbdb1

Array, DIMENSION (p).

Scalar factors of the elementary reflectors that define p₁.

taup2

REAL for sorbdb1

DOUBLE PRECISION for dorbdb1

COMPLEX for cunbdb1

DOUBLE COMPLEX for zunbdb1

Array, DIMENSION (m-p).

Scalar factors of the elementary reflectors that define p₂.

tauq1

REAL for sorbdb1

DOUBLE PRECISION for dorbdb1

COMPLEX for cunbdb1

DOUBLE COMPLEX for zunbdb1

Array, DIMENSION (q).

Scalar factors of the elementary reflectors that define q₁.

info

INTEGER.

= 0: successful exit

< 0: if info = -i, the i-th argument has an illegal value.

Parent topic: LAPACK Auxiliary Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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