SRC/clahr2.f(3)

Okay so first things first: this is the SRC/clahr2.f(3) page from the Library Functions Manual, and it’s giving ✨ auxiliary-routine main character energy ✨ — specifically for CLAHR2.

🌸 NAME 🌸

SRC/clahr2.f

✨ SYNOPSIS ✨

💖 Functions/Subroutines 💖

subroutine clahr2 (n, k, nb, a, lda, tau, t, ldt, y, ldy)

CLAHR2 reduces the specified number of first columns of a general rectangular matrix A so that elements below the specified subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A.

🦋 Function/Subroutine Documentation 🦋

subroutine clahr2 (integer n, integer k, integer nb, complex, dimension( lda, * ) a, integer lda, complex, dimension( nb ) tau, complex, dimension( ldt, nb ) t, integer ldt, complex, dimension( ldy, nb ) y, integer ldy)

CLAHR2 reduces the specified number of first columns of a general rectangular matrix A so that elements below the specified subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A.

✨ Purpose ✨

Okay so here’s the real tea, bestie: CLAHR2 reduces the first NB columns of a complex general n-BY-(n-k+1) matrix A so that elements below the k-th subdiagonal are zero. The reduction is performed by a unitary similarity transformation Q**H * A * Q. The routine returns the matrices V and T which determine Q as a block reflector I - V*T*v**H, and also the matrix Y = A * V * T.

This is an auxiliary routine called by CGEHRD. Periodt. 💅

💅 Parameters 💅

• N — N is INTEGER. The order of the matrix A.
• K — K is INTEGER. The offset for the reduction. Elements below the k-th subdiagonal in the first NB columns are reduced to zero. K < N.
• NB — NB is INTEGER. The number of columns to be reduced.
• A — A is COMPLEX array, dimension (LDA,N-K+1). On entry, the n-by-(n-k+1) general matrix A. On exit, the elements on and above the k-th subdiagonal in the first NB columns are overwritten with the corresponding elements of the reduced matrix; the elements below the k-th subdiagonal, with the array TAU, represent the matrix Q as a product of elementary reflectors. The other columns of A are unchanged. See Further Details.
• LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
• TAU — TAU is COMPLEX array, dimension (NB). The scalar factors of the elementary reflectors. See Further Details.
• T — T is COMPLEX array, dimension (LDT,NB). The upper triangular matrix T.
• LDT — LDT is INTEGER. The leading dimension of the array T. LDT >= NB.
• Y — Y is COMPLEX array, dimension (LDY,NB). The n-by-nb matrix Y.
• LDY — LDY is INTEGER. The leading dimension of the array Y. LDY >= N.

🌟 Author 🌟

• Univ. of Tennessee
• Univ. of California Berkeley
• Univ. of Colorado Denver
• NAG Ltd.

💖 Further Details 💖

The matrix Q is represented as a product of nb elementary reflectors:

Q = H(1) H(2) . . . H(nb).

Each H(i) has the form:

H(i) = I - tau * v * v**H

where tau is a complex scalar, and v is a complex vector with v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in A(i+k+1:n,i), and tau in TAU(i).

The elements of the vectors v together form the (n-k+1)-by-nb matrix V which is needed, with T and Y, to apply the transformation to the unreduced part of the matrix, using an update of the form:

A := (I - V*T*V**H) * (A - Y*V**H).

The contents of A on exit are illustrated by the following example with n = 7, k = 3 and nb = 2:

( a   a   a   a   a )
( a   a   a   a   a )
( a   a   a   a   a )
( h   h   a   a   a )
( v1  h   a   a   a )
( v1  v2  a   a   a )
( v1  v2  a   a   a )

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

This subroutine is a slight modification of LAPACK-3.0's CLAHRD incorporating improvements proposed by Quintana-Orti and Van de Gejin. Note that the entries of A(1:K,2:NB) differ from those returned by the original LAPACK-3.0's CLAHRD routine.

⚠️ Caveat bestie: this subroutine is not backward compatible with LAPACK-3.0's CLAHRD.

📚 References 📚

Gregorio Quintana-Orti and Robert van de Geijn, "Improving the performance of reduction to Hessenberg form," ACM Transactions on Mathematical Software, 32(2):180-194, June 2006.

Definition at line 180 of file clahr2.f.

👑 Author 👑

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