?tpqrt2

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 11/07/2023

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?tpqrt2

Computes a 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.

Syntax

lapack_int LAPACKE_stpqrt2 (int matrix_layout, lapack_int m, lapack_int n, lapack_int l, float * a, lapack_int lda, float * b, lapack_int ldb, float * t, lapack_int ldt);

lapack_int LAPACKE_dtpqrt2 (int matrix_layout, lapack_int m, lapack_int n, lapack_int l, double * a, lapack_int lda, double * b, lapack_int ldb, double * t, lapack_int ldt);

lapack_int LAPACKE_ctpqrt2 (int matrix_layout, lapack_int m, lapack_int n, lapack_int l, lapack_complex_float * a, lapack_int lda, lapack_complex_float * b, lapack_int ldb, lapack_complex_float * t, lapack_int ldt );

lapack_int LAPACKE_ztpqrt2 (int matrix_layout, lapack_int m, lapack_int n, lapack_int l, lapack_complex_double * a, lapack_int lda, lapack_complex_double * b, lapack_int ldb, lapack_complex_double * t, lapack_int ldt );

Include Files

mkl.h

Description

The input matrix C is an (n+m)-by-n matrix

where A is an n-by-n upper triangular matrix, and B is an m-by-n pentagonal matrix consisting of an (m-l)-by-n rectangular matrix B1 on top of an l-by-n upper trapezoidal matrix B2:

The upper trapezoidal matrix B2 consists of the first l rows of an n-by-n upper triangular matrix, where 0 ≤ l ≤ min(m,n). If l=0, B is an m-by-n rectangular matrix. If m=l=n, B is upper triangular. The matrix W contains the elementary reflectors H(i) in the ith column below the diagonal (of A) in the (n+m)-by-n input matrix C so that W can be represented as

Thus, V contains all of the information needed for W, and is returned in array b.

NOTE:

V has the same form as B:

The columns of V represent the vectors which define the H(i)s.

The (m+n)-by-(m+n) block reflector H is then given by

H = I - W*T*W^T for real flavors, and

H = I - W*T*W^H for complex flavors

where W^T is the transpose of W, W^H is the conjugate transpose of W, and T is the upper triangular factor of the block reflector.

Input Parameters

matrix_layout

Specifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR) or column major ( LAPACK_COL_MAJOR ).

m

The total number of rows in the matrix B (m ≥ 0).

n

The number of columns in B and the order of the triangular matrix A (n ≥ 0).

l

The number of rows of the upper trapezoidal part of B (min(m, n) ≥ l ≥ 0).

a, b

Arrays: a, size max(1, lda *n) contains the n-by-n upper triangular matrix A.

b, size max(1,ldb* n) for column major and max(1,ldb*m) for row major, the pentagonal m-by-n matrix B. The first (m-l) rows contain the rectangular B1 matrix, and the next l rows contain the upper trapezoidal B2 matrix.

lda

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

ldb

The leading dimension of b; at least max(1, m) for column major and max(1,n) for row major.

ldt

The leading dimension of t; at least max(1, n).

Output Parameters

a

The elements on and above the diagonal of the array contain the upper triangular matrix R.

b

The pentagonal matrix V.

t

Array, size max(1, ldt *n).

The upper n-by-n upper triangular factor T of the block reflector.

Return Values

This function returns a value info.

If info = 0, the execution is successful.

If info < 0 and info = -i, the ith argument had an illegal value.

If info = -1011, memory allocation error occurred.

Parent topic: LAPACK Auxiliary Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for C

?tpqrt2

Syntax

Include Files

Description

Input Parameters

Output Parameters

Return Values