Visible to Intel only — GUID: GUID-B830AE7B-F650-497E-B4F8-BB7E116650AE
Visible to Intel only — GUID: GUID-B830AE7B-F650-497E-B4F8-BB7E116650AE
?orbdb/?unbdb
Simultaneously bidiagonalizes the blocks of a partitioned orthogonal/unitary matrix.
Syntax
call sorbdb( trans, signs, m, p, q, x11, ldx11, x12, ldx12, x21, ldx21, x22, ldx22, theta, phi, taup1, taup2, tauq1, tauq2, work, lwork, info )
call dorbdb( trans, signs, m, p, q, x11, ldx11, x12, ldx12, x21, ldx21, x22, ldx22, theta, phi, taup1, taup2, tauq1, tauq2, work, lwork, info )
call cunbdb( trans, signs, m, p, q, x11, ldx11, x12, ldx12, x21, ldx21, x22, ldx22, theta, phi, taup1, taup2, tauq1, tauq2, work, lwork, info )
call zunbdb( trans, signs, m, p, q, x11, ldx11, x12, ldx12, x21, ldx21, x22, ldx22, theta, phi, taup1, taup2, tauq1, tauq2, work, lwork, info )
call orbdb( x11,x12,x21,x22,theta,phi,taup1,taup2,tauq1,tauq2[,trans][,signs][,info] )
call unbdb( x11,x12,x21,x22,theta,phi,taup1,taup2,tauq1,tauq2[,trans][,signs][,info] )
Include Files
- mkl.fi, lapack.f90
Description
The routines ?orbdb/?unbdb simultaneously bidiagonalizes the blocks of an m-by-m partitioned orthogonal matrix X:
or unitary matrix:
x11 is p-by-q. q must not be larger than p, m-p, or m-q. Otherwise, x must be transposed and/or permuted in constant time using the trans and signs options.
The orthogonal/unitary matrices p1, p2, q1, and q2 are p-by-p, (m-p)-by-(m-p), q-by-q, (m-q)-by-(m-q), respectively. They are represented implicitly by Housholder vectors.
The bidiagonal matrices b11, b12, b21, and b22 are q-by-q bidiagonal matrices represented implicitly by angles theta(1), ..., theta(q) and phi(1), ..., phi(q-1). b11 and b12 are upper bidiagonal, while b21 and b22 are lower bidiagonal. 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] for details.
p1, p2, q1, and q2 are represented as products of elementary reflectors. .
Input Parameters
- trans
-
CHARACTER
- = 'T':
- x, u1, u2, v1t, v2t are stored in row-major order.
- otherwise
- x, u1, u2, v1t, v2t are stored in column-major order.
- signs
-
CHARACTER
- = 'O':
- The lower-left block is made nonpositive (the "other" convention).
- otherwise
- The upper-right block is made nonpositive (the "default" convention).
- m
-
INTEGER. The number of rows and columns of the matrix X.
- p
-
INTEGER. The number of rows in x11 and x12. 0 ≤p≤m.
- q
-
INTEGER. The number of columns in x11 and x21. 0 ≤q≤ min(p,m-p,m-q).
- x11
-
REAL for sorbdb
DOUBLE PRECISION for dorbdb
COMPLEX for cunbdb
DOUBLE COMPLEX for zunbdb
Array, size (ldx11,*) .
On entry, the top-left block of the orthogonal/unitary matrix to be reduced.
- ldx11
-
INTEGER. The leading dimension of the array X11. If trans = 'T', ldx11≥p. Otherwise, ldx11≥q.
- x12
-
REAL for sorbdb
DOUBLE PRECISION for dorbdb
COMPLEX for cunbdb
DOUBLE COMPLEX for zunbdb
Array, size (ldx12,m-q).
On entry, the top-right block of the orthogonal/unitary matrix to be reduced.
- ldx12
-
INTEGER. The leading dimension of the array X12. If trans = 'N', ldx12≥p. Otherwise, ldx12≥m-q.
- x21
-
REAL for sorbdb
DOUBLE PRECISION for dorbdb
COMPLEX for cunbdb
DOUBLE COMPLEX for zunbdb
Array, size (ldx21,q).
On entry, the bottom-left block of the orthogonal/unitary matrix to be reduced.
- ldx21
-
INTEGER. The leading dimension of the array X21. If trans = 'N', ldx21≥m-p. Otherwise, ldx21≥q.
- x22
-
REAL for sorbdb
DOUBLE PRECISION for dorbdb
COMPLEX for cunbdb
DOUBLE COMPLEX for zunbdb
Array, size (ldx22,m-q).
On entry, the bottom-right block of the orthogonal/unitary matrix to be reduced.
- ldx22
-
INTEGER. The leading dimension of the array X21. If trans = 'N', ldx22≥m-p. Otherwise, ldx22≥m-q.
- work
-
REAL for sorbdb
DOUBLE PRECISION for dorbdb
COMPLEX for cunbdb
DOUBLE COMPLEX for zunbdb
Workspace array, size (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
-
On exit, the form depends on trans:
- If trans='N',
- the columns of the lower triangle of x11 specify reflectors for p1, the rows of the upper triangle of x11(1:q - 1, q:q - 1) specify reflectors for q1
- otherwise trans='T',
- the rows of the upper triangle of x11 specify reflectors for p1, the columns of the lower triangle of x11(1:q - 1, q:q - 1) specify reflectors for q1
- x12
-
On exit, the form depends on trans:
- If trans='N',
- the columns of the upper triangle of x12 specify the first p reflectors for q2
- otherwise trans='T',
- the columns of the lower triangle of x12 specify the first p reflectors for q2
- x21
-
On exit, the form depends on trans:
- If trans='N',
- the columns of the lower triangle of x21 specify the reflectors for p2
- otherwise trans='T',
- the columns of the upper triangle of x21 specify the reflectors for p2
- x22
-
On exit, the form depends on trans:
- If trans='N',
- the rows of the upper triangle of x22(q+1:m-p,p+1:m-q) specify the last m-p-q reflectors for q2
- otherwise trans='T',
- the columns of the lower triangle of x22(p+1:m-q,q+1:m-p) specify the last m-p-q reflectors for p2
- theta
-
REAL for sorbdb
DOUBLE PRECISION for dorbdb
COMPLEX for cunbdb
DOUBLE COMPLEX for zunbdb
Array, size (q). The entries of bidiagonal blocks b11, b12, b21, and b22 can be computed from the angles theta and phi. See the Description section for details.
- phi
-
REAL for sorbdb
DOUBLE PRECISION for dorbdb
COMPLEX for cunbdb
DOUBLE COMPLEX for zunbdb
Array, size (q-1). The entries of bidiagonal blocks b11, b12, b21, and b22 can be computed from the angles theta and phi. See the Description section for details.
- taup1
-
REAL for sorbdb
DOUBLE PRECISION for dorbdb
COMPLEX for cunbdb
DOUBLE COMPLEX for zunbdb
Array, size (p).
Scalar factors of the elementary reflectors that define p1. - taup2
-
REAL for sorbdb
DOUBLE PRECISION for dorbdb
COMPLEX for cunbdb
DOUBLE COMPLEX for zunbdb
Array, size (m-p).
Scalar factors of the elementary reflectors that define p2. - tauq1
-
REAL for sorbdb
DOUBLE PRECISION for dorbdb
COMPLEX for cunbdb
DOUBLE COMPLEX for zunbdb
Array, size (q).
Scalar factors of the elementary reflectors that define q1. - tauq2
-
REAL for sorbdb
DOUBLE PRECISION for dorbdb
COMPLEX for cunbdb
DOUBLE COMPLEX for zunbdb
Array, size (m-q).
Scalar factors of the elementary reflectors that define q2. - info
-
INTEGER.
= 0: successful exit
< 0: if info = -i, the i-th argument has 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 reconstructible arguments, see Fortran 95 Interface Conventions.
Specific details for the routine ?orbdb/?unbdb interface are as follows:
- x11
-
Holds the block of matrix X of size (p, q).
- x12
-
Holds the block of matrix X of size (p, m-q).
- x21
-
Holds the block of matrix X of size (m-p, q).
- x22
-
Holds the block of matrix X of size (m-p, m-q).
- theta
-
Holds the vector of length q.
- phi
-
Holds the vector of length q-1.
- taup1
-
Holds the vector of length p.
- taup2
-
Holds the vector of length m-p.
- tauq1
-
Holds the vector of length q.
- taupq2
-
Holds the vector of length m-q.
- trans
-
Must be 'N' or 'T'.
- signs
-
Must be 'O' or 'D'.