# sorbdb4.f - Man Page

SRC/sorbdb4.f

## Synopsis

### Functions/Subroutines

subroutine **sorbdb4** (m, p, q, x11, ldx11, x21, ldx21, theta, phi, taup1, taup2, tauq1, phantom, work, lwork, info)**SORBDB4**

## Function/Subroutine Documentation

### subroutine sorbdb4 (integer m, integer p, integer q, real, dimension(ldx11,*) x11, integer ldx11, real, dimension(ldx21,*) x21, integer ldx21, real, dimension(*) theta, real, dimension(*) phi, real, dimension(*) taup1, real, dimension(*) taup2, real, dimension(*) tauq1, real, dimension(*) phantom, real, dimension(*) work, integer lwork, integer info)

**SORBDB4**

**Purpose:**

SORBDB4 simultaneously bidiagonalizes the blocks of a tall and skinny matrix X with orthonormal columns: [ B11 ] [ X11 ] [ P1 | ] [ 0 ] [-----] = [---------] [-----] Q1**T . [ X21 ] [ | P2 ] [ B21 ] [ 0 ] X11 is P-by-Q, and X21 is (M-P)-by-Q. M-Q must be no larger than P, M-P, or Q. Routines SORBDB1, SORBDB2, and SORBDB3 handle cases in which M-Q is not the minimum dimension. The orthogonal matrices P1, P2, and Q1 are P-by-P, (M-P)-by-(M-P), and (M-Q)-by-(M-Q), respectively. They are represented implicitly by Householder vectors. B11 and B12 are (M-Q)-by-(M-Q) bidiagonal matrices represented implicitly by angles THETA, PHI.

**Parameters***M*M is INTEGER The number of rows X11 plus the number of rows in X21.

*P*P is INTEGER The number of rows in X11. 0 <= P <= M.

*Q*Q is INTEGER The number of columns in X11 and X21. 0 <= Q <= M and M-Q <= min(P,M-P,Q).

*X11*X11 is REAL array, dimension (LDX11,Q) On entry, the top block of the matrix X to be reduced. On exit, the columns of tril(X11) specify reflectors for P1 and the rows of triu(X11,1) specify reflectors for Q1.

*LDX11*LDX11 is INTEGER The leading dimension of X11. LDX11 >= P.

*X21*X21 is REAL array, dimension (LDX21,Q) On entry, the bottom block of the matrix X to be reduced. On exit, the columns of tril(X21) specify reflectors for P2.

*LDX21*LDX21 is INTEGER The leading dimension of X21. LDX21 >= M-P.

*THETA*THETA is REAL array, dimension (Q) The entries of the bidiagonal blocks B11, B21 are defined by THETA and PHI. See Further Details.

*PHI*PHI is REAL array, dimension (Q-1) The entries of the bidiagonal blocks B11, B21 are defined by THETA and PHI. See Further Details.

*TAUP1*TAUP1 is REAL array, dimension (M-Q) The scalar factors of the elementary reflectors that define P1.

*TAUP2*TAUP2 is REAL array, dimension (M-Q) The scalar factors of the elementary reflectors that define P2.

*TAUQ1*TAUQ1 is REAL array, dimension (Q) The scalar factors of the elementary reflectors that define Q1.

*PHANTOM*PHANTOM is REAL array, dimension (M) The routine computes an M-by-1 column vector Y that is orthogonal to the columns of [ X11; X21 ]. PHANTOM(1:P) and PHANTOM(P+1:M) contain Householder vectors for Y(1:P) and Y(P+1:M), respectively.

*WORK*WORK is REAL array, dimension (LWORK)

*LWORK*LWORK is INTEGER The dimension of the array WORK. 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.

*INFO*INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value.

**Author**Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

The upper-bidiagonal blocks B11, B21 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 a THETA with a sine or cosine of a PHI. See [1] or SORCSD for details. P1, P2, and Q1 are represented as products of elementary reflectors. See SORCSD2BY1 for details on generating P1, P2, and Q1 using SORGQR and SORGLQ.

**References:**

[1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms, 50(1):33-65, 2009.

Definition at line **211** of file **sorbdb4.f**.

## Author

Generated automatically by Doxygen for LAPACK from the source code.

## Referenced By

The man page sorbdb4(3) is an alias of sorbdb4.f(3).

Tue Nov 28 2023 12:08:42 Version 3.12.0 LAPACK