?gehd2

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 4/28/2026

Version

Public

?gehd2

Reduces a general square matrix to upper Hessenberg form using an unblocked algorithm.

Syntax

call sgehd2 ( n , ilo , ihi , a , lda , tau , work , info )

call dgehd2 ( n , ilo , ihi , a , lda , tau , work , info )

call cgehd2 ( n , ilo , ihi , a , lda , tau , work , info )

call zgehd2 ( n , ilo , ihi , a , lda , tau , work , info )

Include Files

mkl.fi

Description

sgehd2 dgehd2 cgehd2 zgehd2

The routine reduces a real/complex general matrix A to upper Hessenberg form H by an orthogonal or unitary similarity transformation Q^{T}*A*Q = H (for real flavors) or Q^{H}*A*Q = H (for complex flavors).

The routine does not form the matrix Q explicitly. Instead, Q is represented as a product of elementary reflectors .

Input Parameters

n: INTEGER The order of the matrix A ( n≥ 0 ).
ilo , ihi: INTEGER . It is assumed that A is already upper triangular in rows and columns 1:ilo -1 and ihi+1:n . If A has been output by ?gebal , then ilo and ihi must contain the values returned by that routine. Otherwise they should be set to ilo = 1 and ihi = n . Constraint: 1 ≤ ilo ≤ ihi ≤ max(1, n) .
a , work: REAL for sgehd2 DOUBLE PRECISION for dgehd2 COMPLEX for cgehd2 DOUBLE COMPLEX for zgehd2 . Arrays: a (lda,*) contains the n -by- n matrix A to be reduced. The second dimension of a must be at least max(1, n) . work(n) is a workspace array.
lda: INTEGER . The leading dimension of a ; at least max(1, n) .

Output Parameters

a: On exit, the upper triangle and the first subdiagonal of A are overwritten with the upper Hessenberg matrix H and the elements below the first subdiagonal, with the array tau , represent the orthogonal/unitary matrix Q as a product of elementary reflectors. See Application Notes below.
tau: REAL for sgehd2 DOUBLE PRECISION for dgehd2 COMPLEX for cgehd2 DOUBLE COMPLEX for zgehd2 . Array, DIMENSION at least max (1, n-1) . Contains the scalar factors of elementary reflectors. See Application Notes below.

info

INTEGER .

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

Application Notes

The matrix \(Q\) is represented as a product of ( ihi - ilo ) elementary reflectors

\(Q = H(\text{ilo}) H(\text{ilo}+1) \cdots H(\text{ihi}-1)\)

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, and v is a real/complex vector with v(1:i) = 0 , v(i+1) = 1 and v(ihi+1:n) = 0 .

On exit, v(i+2:ihi) is stored in a(i+2:ihi, i) and tau in tau(i) .

The contents of a are illustrated by the following example, with n = 7 , ilo = 2 and ihi = 6 :

where a denotes an element of the original matrix \(A\) , h denotes a modified element of the upper Hessenberg matrix \(H\) , and v_i denotes an element of the vector defining \(H\) (i).

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

?gehd2

Syntax

Include Files

Description

Input Parameters

Output Parameters

Application Notes