?tpqrt

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 4/28/2026

Version

Public

?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.

Syntax

call stpqrt ( m , n , l , nb , a , lda , b , ldb , t , ldt , work , info )

call dtpqrt ( m , n , l , nb , a , lda , b , ldb , t , ldt , work , info )

call ctpqrt ( m , n , l , nb , a , lda , b , ldb , t , ldt , work , info )

call ztpqrt ( m , n , l , nb , a , lda , b , ldb , t , ldt , work , info )

call tpqrt ( a , b , t , nb [ , info ] )

Include Files

mkl.fi , mkl_lapack.f90

Description

stpqrt dtpqrt ctpqrt ztpqrt tpqrt

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 elementary reflectors H(i) are stored in the i th column below the diagonal in the ( n + m )-by- n input matrix C . The structure of vectors defining the elementary reflectors is illustrated by:

The elements of the unit matrix I are not stored. Thus, V contains all of the necessary information, and is returned in array b .

NOTE:

Note that V has the same form as B :

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

The number of blocks is k = ceiling( n / nb ), where each block is of order nb except for the last block, which is of order ib = n - ( k -1)* nb . For each of the k blocks, an upper triangular block reflector factor is computed: T1 , T2 , …, Tk . The nb -by- nb ( ib -by- ib for the last block) Ti s are stored in the nb -by- n array t as

t = [T1T2 ... Tk] .

Input Parameters

m

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

n

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

l

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

nb

INTEGER . The block size to use in the blocked QR factorization ( n ≥ nb ≥ 1).

a , b , work

REAL for stpqrt DOUBLE PRECISION for dtpqrt COMPLEX for ctpqrt COMPLEX*16 for ztpqrt .

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

b size ( ldb , n ) , 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.

work size ( nb , n ) is a workspace array.

lda

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

ldb

INTEGER . The leading dimension of b ; at least max(1, m ) .

ldt

INTEGER . The leading dimension of t ; at least nb .

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

REAL for stpqrt DOUBLE PRECISION for dtpqrt COMPLEX for ctpqrt COMPLEX*16 for ztpqrt .

Array, size ( ldt , n ) .

The upper triangular block reflectors stored in compact form as a sequence of upper triangular blocks.

info

INTEGER .

If info = 0, the execution is successful.
If info < 0 and info = -i , the i th argument had an illegal value.

Return Values

No return value, info is an Output Parameter.

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

?tpqrt

Syntax

Include Files

Description

Input Parameters

Output Parameters

Return Values