Developer Reference for Intel® oneAPI Math Kernel Library for Fortran
?latsqr
Computes a blocked Tall-Skinny QR matrix factorization.
Syntax
call slatsqr ( m , n , mb , nb , a , lda , t , ldt , work , lwork , info )
call dlatsqr ( m , n , mb , nb , a , lda , t , ldt , work , lwork , info )
call clatsqr ( m , n , mb , nb , a , lda , t , ldt , work , lwork , info )
call zlatsqr ( m , n , mb , nb , a , lda , t , ldt , work , lwork , info )
Description
?latsqr computes a blocked Tall-Skinny QR (TSQR) factorization of an m -by- n matrix A , where m ≥ n : A = Q * R .
TSQR performs QR by a sequence of orthogonal transformations, representing Q as a product of other orthogonal matrices
Q = Q (1) * Q (2) * … * Q ( k )
where each Q ( i ) zeros out subdiagonal entries of a block of mb rows of A :
Q (1) zeros out the subdiagonal entries of rows 1:MB of A ,
Q (2) zeros out the bottom mb - n rows of rows [1: n , mb + 1:2* mb - n ] of A ,
Q (3) zeros out the bottom mb - n rows of rows [1: n , 2* mb - n + 1:3* mb - 2* n ] of A … .
Q (1) is computed by geqrt , which represents Q (1) by Householder vectors stored under the diagonal of rows 1: mb of a , and by upper triangular block reflectors, stored in array t (1: ldt , 1: n ). For more information see geqrt (Computes a blocked QR factorization of a general real or complex matrix using the compact WY representation of Q.) .
Q ( i ) for i > 1 is computed by tpqrt , which represents Q ( i ) by Householder vectors stored in rows [( i - 1)*( mb - n ) + n + 1: i *( mb - n ) + n ] of a , and by upper triangular block reflectors, stored in array t (1: ldt , ( i - 1)* n + 1: i * n ). The last Q ( k ) may use fewer rows. For more information, see tpqrt (Computes a blocked QR factorization of a real or complex “triangular-pentagonal” matrix, which is composed of a triangular block and a pentagonal block, using the compact WY representation for Q.) . For more details of the overall algorithm, see [DEMMEL12].
Input Parameters
m
INTEGER . The number of rows of the matrix A . m ≥ 0.
n
INTEGER . The number of columns of the matrix A . m ≥ n ≥ 0.
mb
INTEGER . The row block size to be used in the blocked QR. mb > n .
nb
INTEGER . The column block size to be used in the blocked QR. n ≥ nb ≥ 1.
- a
-
REAL for slatsqr DOUBLE PRECISION for dlatsqr COMPLEX for clatsqr COMPLEX*16 for zlatsqr Array of size ( lda , n ). On entry, the m -by- n matrix A .
lda
INTEGER . The leading dimension of the array a . lda ≥ max(1, m ).
ldt
INTEGER . The leading dimension of the array t . ldt ≥ nb .
lwork
INTEGER . The dimension of the array work . lwork ≥ nb * 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 .
Output Parameters
- a
-
On exit, the elements on and above the diagonal of the array contain the n -by- n upper triangular matrix R and the elements below the diagonal represent Q by the columns of blocked V .
- t
-
REAL for slatsqr DOUBLE PRECISION for dlatsqr COMPLEX for clatsqr COMPLEX*16 for zlatsqr Array of size ( ldt , n * Number_of_row_blocks ) where Number_of_row_blocks = ceiling(( m - n )/( mb - n )) The blocked upper triangular block reflectors stored in compact form as a sequence of upper triangular blocks.
- work
-
REAL for slatsqr DOUBLE PRECISION for dlatsqr COMPLEX for clatsqr COMPLEX*16 for zlatsqr Workspace array of size (max(1, lwork )).
info
INTEGER .
info = 0: successful exit.
info < 0: if info = - i , the i -th argument had an illegal value.