DSYT21(3)

💅 NAME

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/TESTING/EIG/dsyt21.f

yeah bestie that's the whole name, it's a whole ✨path✨, it's giving "my parents named me after the street I was born on" 😭 — this is dsyt21.f from the LAPACK TESTING/EIG directory, and it does not do short names 💀

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🦋 SYNOPSIS

Functions/Subroutines

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

DSYT21 — that's it, that's the function 🎀

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🌸 Function/Subroutine Documentation

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:

ok so listen, this one is a checker. Like, it doesn't compute the decomposition, it vibe checks it. periodt.

!>
!> 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).
!>
!> 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).
!>

🙈 lowkey tea: in that one line about "stored in the array and whose scaling constants are in ." the names literally got eaten by the doc generator. the blanks are canon, I'm not inventing anything, bestie 🫶

---

✨ 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 )
!>

(and yes it says "Housholder" in the source, we respect the typo, it's iconic 💅)

🧚 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.
!>

⚠️ bestie read the note: with ITYPE=2 or 3, V gets modified and restored — that sub/superdiagonal goes to one and comes back. it's giving "borrowed your hoodie and put it back, slightly different" 😭 don't panic, it's documented behaviour, not chaos 💀

🌟 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)
!>

that's 2*N**2, no cap, it's a big girl array 🔥

⭐ 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.
!>

so RESULT(1) always gets written, RESULT(2) only when ITYPE=1 — the other two slots get left alone, main character energy for RESULT(1) only 🎀

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🌈 Author

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

Definition at line 205 of file dsyt21.f.

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💜 Author

Generated automatically by Doxygen for LAPACK from the source code.

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and that's the whole page bestie, no crumbs left 🫶💖 — this gorgeous thing came to you courtesy of LAPACK, Version 3.12.0, and the header path /home/abui...IG/dsyt21.f(3) was right up there at the top the whole time. slay, go test some symmetric decompositions ✨👑