SRC/chptrd.f(3)

SRC/chptrd.f(3) — Library Functions Manual vibes 💅

✨ NAME ✨

SRC/chptrd.f

🌸 SYNOPSIS 🌸

Functions/Subroutines

subroutine chptrd (uplo, n, ap, d, e, tau, info)
    CHPTRD

🛠️ Function/Subroutine Documentation 🛠️

subroutine chptrd (character uplo, integer n, complex, dimension( * ) ap,
    real, dimension( * ) d, real, dimension( * ) e, complex, dimension( * )
    tau, integer info)
    CHPTRD

🎯 Purpose

okay so bestie, CHPTRD is out here reducing a complex Hermitian matrix A stored in packed form to real symmetric tridiagonal form T by a unitary similarity transformation: Q**H * A * Q = T. iconic behavior, honestly. 💜

📦 Parameters

• UPLO — UPLO is CHARACTER*1; = 'U': upper triangle of A is stored; = 'L': lower triangle of A is stored.
• N — N is INTEGER; the order of the matrix A. N >= 0.
• AP — AP is COMPLEX array, dimension (N*(N+1)/2). On entry, it holds the upper or lower triangle of the Hermitian matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows:
  • if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
  • if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.

On exit, if UPLO = 'U', the diagonal and first superdiagonal of A are overwritten by the corresponding elements of the tridiagonal matrix T, and the elements above the first superdiagonal, with the array TAU, represent the unitary matrix Q as a product of elementary reflectors. If UPLO = 'L', the diagonal and first subdiagonal of A are over-written by the corresponding elements of the tridiagonal matrix T, and the elements below the first subdiagonal, with the array TAU, represent the unitary matrix Q as a product of elementary reflectors. See Further Details for the lore.

• D — D is REAL array, dimension (N); the diagonal elements of the tridiagonal matrix T: D(i) = A(i,i).
• E — E is REAL array, dimension (N-1); the off-diagonal elements of the tridiagonal matrix T: E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
• TAU — TAU is COMPLEX array, dimension (N-1); the scalar factors of the elementary reflectors (see Further Details).
• INFO — INFO is INTEGER; = 0: successful exit; < 0: if INFO = -i, the i-th argument had an illegal value.

👑 Author

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

🔍 Further Details

If UPLO = 'U', the matrix Q is represented as a product of elementary reflectors:

Q = H(n-1) . . . H(2) H(1).

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(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in AP, overwriting A(1:i-1,i+1), and tau is stored in TAU(i).

If UPLO = 'L', the matrix Q is represented as a product of elementary reflectors:

Q = H(1) H(2) . . . H(n-1).

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) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in AP, overwriting A(i+2:n,i), and tau is stored in TAU(i). ✨

📍 Definition at line 150 of file chptrd.f.

This whole function definition lives at line 150 of chptrd.f. noted, bestie 🫶

🧚 Author 🧚

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that's the page, bestie! The footer cuties are LAPACK Version 3.12.0 and SRC/chptrd.f(3) — no date line on this one, just those source/version details. 💖✨