DLAHR2(3)

bestie, welcome to DLAHR2(3) — it's giving Library Functions Manual, section 3, and we're about to make this LAPACK subroutine make sense. 💅

💖 NAME 💖

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/dlahr2.f

That's the whole name/address of this file, no cap. 🫶

✨ SYNOPSIS ✨

🧚 Functions/Subroutines

       subroutine DLAHR2 (n, k, nb, a, lda, tau, t, ldt, y, ldy)
           DLAHR2 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 DLAHR2 (integer n, integer k, integer nb, double precision,
       dimension( lda, * ) a, integer lda, double precision, dimension( nb )
       tau, double precision, dimension( ldt, nb ) t, integer ldt, double
       precision, dimension( ldy, nb ) y, integer ldy)

DLAHR2 is the main character here: it takes the first columns you specify of a general rectangular matrix A, zeros out everything below the chosen subdiagonal, and hands back the auxiliary matrices you need to apply that transformation to the rest of A. 💅

🎯 Purpose

ok so the whole point, bestie: DLAHR2 reduces the first NB columns of A, a real general n-by-(n-k+1) matrix, so that elements below the k-th subdiagonal are zero. The reduction is performed by an orthogonal similarity transformation Q**T * A * Q.

The routine returns the matrices V and T which determine Q as a block reflector I - V*T*V**T, and also the matrix Y = A * V * T.

Lowkey iconic: this is an auxiliary routine called by DGEHRD. 🦋

🌸 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (NB). The scalar factors of the elementary reflectors. See Further Details.
• T — T is DOUBLE PRECISION 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 DOUBLE PRECISION 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

ok bestie, here's the math tea. 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**T

where tau is a real scalar, and v is a real 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**T) * (A - Y*V**T).

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).

⚠️ Not backward compatible alert: this subroutine is a slight modification of LAPACK-3.0's DLAHRD 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 DLAHRD routine. (This subroutine is not backward compatible with LAPACK-3.0's DLAHRD.)

📚 References:

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

Definition at line 180 of file dlahr2.f.

🤖 Author

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that's the page, bestie 💖 LAPACK Version 3.12.0, footer source path /home/abui...RC/dlahr2.f(3), and no separate date is printed in that footer — so I'm not inventing one. Periodt. 💅✨