SRC/DEPRECATED/slahrd.f(3)

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🌸 NAME

SRC/DEPRECATED/slahrd.f

✨ SYNOPSIS

💅 Functions/Subroutines

• subroutine slahrd (n, k, nb, a, lda, tau, t, ldt, y, ldy) — SLAHRD reduces the first nb columns of a general rectangular matrix A so that elements below the k-th subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A. 🦋

🛠️ FUNCTION/SUBROUTINE DOCUMENTATION

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

SLAHRD reduces the first nb columns of a general rectangular matrix A so that elements below the k-th subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A. 💖

🎯 PURPOSE

⚠️ **Deprecated alert: this routine is deprecated and has been replaced by routine SLAHR2. Periodt. 💀

SLAHRD reduces the first NB columns of a real general n-by-(n-k+1) matrix A 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. ✨

🧩 PARAMETERS

• N — N is INTEGER; it’s 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. 💅
• NB — NB is INTEGER; the number of columns to be reduced. 🎀
• A — A is REAL 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 REAL array, dimension (NB); the scalar factors of the elementary reflectors. See Further Details. 💖
• T — T is REAL 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 REAL 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**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   h   a   a   a )
( a   h   a   a   a )
( a   h   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). 🌸

Definition at line 166 of file slahrd.f.

🤖 AUTHOR

Generated automatically by Doxygen for LAPACK from the source code.

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