?gerq2

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 4/28/2026

Version

Public

Computes the RQ factorization of a general rectangular matrix using an unblocked algorithm.

call sgerq2 ( m , n , a , lda , tau , work , info )

call dgerq2 ( m , n , a , lda , tau , work , info )

call cgerq2 ( m , n , a , lda , tau , work , info )

call zgerq2 ( m , n , a , lda , tau , work , info )

sgerq2 dgerq2 cgerq2 zgerq2

The routine computes a RQ factorization of a real/complex m -by- n matrix A as A = R*Q .

The routine does not form the matrix Q explicitly. Instead, Q is represented as a product of min( m , n ) elementary reflectors :

Q = H(1)*H(2)* ... *H(k) for real flavors, or

Q = H(1)^{H}*H(2)^{H}* ... *H(k)^{H} for complex flavors

where k = min(m, n).

Each H (i) has the form

H(i) = I - tau*v*v^{T} for real flavors, or

H(i) = I - tau*v*v^{H} for complex flavors

where tau is a real/complex scalar stored in tau (i), and v is a real/complex vector with v(n-k+i+1:n) = 0 and v(n-k+i) = 1 .

On exit, v(1:n-k+i-1) is stored in a(m-k+i, 1:n-k+i-1) .

m: INTEGER . The number of rows in the matrix A ( m≥ 0 ).
n: INTEGER . The number of columns in A ( n≥ 0 ).
a , work: REAL for sgerq2 DOUBLE PRECISION for dgerq2 COMPLEX for cgerq2 DOUBLE COMPLEX for zgerq2 . Arrays: a(lda,*) contains the m -by- n matrix A . The second dimension of a must be at least max(1, n) . work(m) is a workspace array.
lda: INTEGER . The leading dimension of a ; at least max(1, m) .

a: Overwritten by the factorization data as follows: on exit, if m ≤ n , the upper triangle of the subarray a(1:m, n-m+1:n ) contains the m -by- m upper triangular matrix R ; if m > n , the elements on and above the ( m - n )-th subdiagonal contain the m -by- n upper trapezoidal matrix R ; the remaining elements, with the array tau , represent the orthogonal/unitary matrix Q as a product of elementary reflectors.
tau: REAL for sgerq2 DOUBLE PRECISION for dgerq2 COMPLEX for cgerq2 DOUBLE COMPLEX for zgerq2 . Array, DIMENSION at least max(1, min(m, n)) . Contains scalar factors of the elementary reflectors.

info

INTEGER .