TESTING/EIG/dsyt21.f(3)

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version 3.12.1 · 2 versions
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💖 NAME 💖

okay so bestie, welcome to TESTING/EIG/dsyt21.f(3) from the Library Functions Manual 🫶 the actual name is TESTING/EIG/dsyt21.f, and this whole page is about the DSYT21 checker 💅

✨ SYNOPSIS ✨

Functions/Subroutines

  • subroutine dsyt21 (itype, uplo, n, kband, a, lda, d, e, u, ldu, v, ldv, tau, work, result) — DSYT21 👑
subroutine dsyt21 (itype, uplo, n, kband, a, lda, d, e, u, ldu, v, ldv,
    tau, work, result)

🦋 Function/Subroutine Documentation 🦋

subroutine dsyt21 (...)

subroutine dsyt21 (integer itype, character uplo, integer n, integer kband,
    double precision, dimension( lda, * ) a, integer lda, double precision,
    dimension( * ) d, double precision, dimension( * ) e, double precision,
    dimension( ldu, * ) u, integer ldu, double precision, dimension( ldv, *
    ) v, integer ldv, double precision, dimension( * ) tau, double
    precision, dimension( * ) work, double precision, dimension( 2 ) result)

DSYT21 💖

Purpose

DSYT21 generally checks a decomposition of the form

A = U S U**T

where **T means transpose, A is symmetric, U is orthogonal, and S is diagonal (if KBAND=0) or symmetric tridiagonal (if KBAND=1). 🎀

If ITYPE=1, then U is represented as a dense matrix; otherwise U is expressed as a product of Householder transformations, whose vectors are stored in the array and whose scaling constants are in . We shall use the letter to refer to the product of Householder transformations (which should be equal to U). (the original has blanks right there, so I’m leaving them blank, iykyk 😭)

Specifically, if ITYPE=1, then:

RESULT(1) = | A - U S U**T | / ( |A| n ulp ) and
RESULT(2) = | I - U U**T | / ( n ulp )

If ITYPE=2, then:

RESULT(1) = | A - V S V**T | / ( |A| n ulp )

If ITYPE=3, then:

RESULT(1) = | I - V U**T | / ( n ulp )

For ITYPE > 1, the transformation U is expressed as a product

V = H(1)...H(n-2),  where H(j) = I  -  tau(j) v(j) v(j)**T

and each vector v(j) has its first j elements 0 and the remaining n-j elements stored in V(j+1:n,j). 💀 slay math, no cap

Parameters

  • ITYPE — ITYPE is INTEGER. Specifies the type of tests to be performed. 👑
    • 1: U expressed as a dense orthogonal matrix:
RESULT(1) = | A - U S U**T | / ( |A| n ulp )  and
RESULT(2) = | I - U U**T | / ( n ulp )
  • 2: U expressed as a product V of Housholder transformations:
RESULT(1) = | A - V S V**T | / ( |A| n ulp )
  • 3: U expressed both as a dense orthogonal matrix and as a product of Housholder transformations:
RESULT(1) = | I - V U**T | / ( n ulp )
  • UPLO — UPLO is CHARACTER. If UPLO='U', the upper triangle of A and V will be used and the (strictly) lower triangle will not be referenced. If UPLO='L', the lower triangle of A and V will be used and the (strictly) upper triangle will not be referenced. 🌸
  • N — N is INTEGER. The size of the matrix. If it is zero, DSYT21 does nothing. It must be at least zero. ✨
  • KBAND — KBAND is INTEGER. The bandwidth of the matrix. It may only be zero or one. If zero, then S is diagonal, and E is not referenced. If one, then S is symmetric tri-diagonal. 🎀
  • A — A is DOUBLE PRECISION array, dimension (LDA, N). The original (unfactored) matrix. It is assumed to be symmetric, and only the upper (UPLO='U') or only the lower (UPLO='L') will be referenced. 👑
  • LDA — LDA is INTEGER. The leading dimension of A. It must be at least 1 and at least N. 💅
  • D — D is DOUBLE PRECISION array, dimension (N). The diagonal of the (symmetric tri-) diagonal matrix. 🍓
  • E — E is DOUBLE PRECISION array, dimension (N-1). The off-diagonal of the (symmetric tri-) diagonal matrix. E(1) is the (1,2) and (2,1) element, E(2) is the (2,3) and (3,2) element, etc. Not referenced if KBAND=0. 🦋
  • U — U is DOUBLE PRECISION array, dimension (LDU, N). If ITYPE=1 or 3, this contains the orthogonal matrix in the decomposition, expressed as a dense matrix. If ITYPE=2, then it is not referenced. 🫶
  • LDU — LDU is INTEGER. The leading dimension of U. LDU must be at least N and at least 1. 🌈
  • V — V is DOUBLE PRECISION array, dimension (LDV, N). If ITYPE=2 or 3, the columns of this array contain the Householder vectors used to describe the orthogonal matrix in the decomposition. If UPLO='L', then the vectors are in the lower triangle, if UPLO='U', then in the upper triangle. NOTE If ITYPE=2 or 3, V is modified and restored. The subdiagonal (if UPLO='L') or the superdiagonal (if UPLO='U') is set to one, and later reset to its original value, during the course of the calculation. If ITYPE=1, then it is neither referenced nor modified. 😭 important, bestie!
  • LDV — LDV is INTEGER. The leading dimension of V. LDV must be at least N and at least 1. 💜
  • TAU — TAU is DOUBLE PRECISION array, dimension (N). If ITYPE >= 2, then TAU(j) is the scalar factor of v(j) v(j)**T in the Householder transformation H(j) of the product U = H(1)...H(n-2) If ITYPE < 2, then TAU is not referenced. 🔥
  • WORK — WORK is DOUBLE PRECISION array, dimension (2*N**2). 🧚
  • RESULT — RESULT is DOUBLE PRECISION array, dimension (2). The values computed by the two tests described above. The values are currently limited to 1/ulp, to avoid overflow. RESULT(1) is always modified. RESULT(2) is modified only if ITYPE=1. 👑

Author

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

Definition

Definition at line 205 of file dsyt21.f. 📌

🌟 Author 🌟

Generated automatically by Doxygen for LAPACK from the source code. 💖

okay that’s the whole glow-up 💅 the source footer line is LAPACK Version 3.12.0 TESTING/EIG/dsyt21.f(3), and there’s no separate date shown in the original footer, so I’m not inventing one — periodt ✨🫶

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