DSYT22(3)

💅 DSYT22(3) — Library Functions Manual

okay bestie, welcome to the manual page for DSYT22(3). It lives in the Library Functions Manual, and the NAME section is literally this file path:

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/TESTING/EIG/dsyt22.f

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🦋 SYNOPSIS

✨ Functions/Subroutines

subroutine DSYT22 (itype, uplo, n, m, kband, a, lda, d, e, u, ldu, v,
    ldv, tau, work, result)
    DSYT22

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🌸 Function/Subroutine Documentation

Here's the full signature, copied exactly because we do not play with source code around here 💅:

subroutine DSYT22 (integer itype, character uplo, integer n, integer m,
    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)
    DSYT22

✨ Purpose

So, the vibe is that DSYT22 generally checks a decomposition of this form:

A U = U S

where A is symmetric, the columns of U are orthonormal, 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 the U is expressed as a product of Householder transformations, whose vectors are stored in the array and whose scaling constants are in — then the source text has a lil glitch and literally says we shall use the letter — and then to refer to the product of Householder transformations (which should be equal to U). 🫠 I'm not inventing the missing bits, bestie, that's genuinely what the page says.

Specifically, if ITYPE=1, then the page gives:

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

Then the ITYPE argument description hits us with these formulas for 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 )

Now the arguments, one by one, no skips, no cap:

• ITYPE — 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 ).
• UPLO — CHARACTER. If UPLO='U', the upper triangle of A will be used and the (strictly) lower triangle will not be referenced. If UPLO='L', the lower triangle of A will be used and the (strictly) upper triangle will not be referenced. Not modified.
• N — INTEGER. The size of the matrix. If it is zero, DSYT22 does nothing. It must be at least zero. Not modified.
• M — INTEGER. The number of columns of U. If it is zero, DSYT22 does nothing. It must be at least zero. Not modified.
• KBAND — 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. Not modified.
• A — 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. Not modified.
• LDA — INTEGER. The leading dimension of A. It must be at least 1 and at least N. Not modified.
• D — DOUBLE PRECISION array, dimension (N). The diagonal of the (symmetric tri-) diagonal matrix. Not modified.
• E — DOUBLE PRECISION array, dimension (N). The off-diagonal of the (symmetric tri-) diagonal matrix. E(1) is ignored, E(2) is the (1,2) and (2,1) element, etc. Not referenced if KBAND=0. Not modified.
• U — 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. Not modified.
• LDU — INTEGER. The leading dimension of U. LDU must be at least N and at least 1. Not modified.
• V — DOUBLE PRECISION array, dimension (LDV, N). If ITYPE=2 or 3, the lower triangle of this array contains the Householder vectors used to describe the orthogonal matrix in the decomposition. If ITYPE=1, then it is not referenced. Not modified.
• LDV — INTEGER. The leading dimension of V. LDV must be at least N and at least 1. Not modified.
• TAU — 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. Not modified.
• WORK — DOUBLE PRECISION array, dimension (2*N**2). Workspace. Modified.
• RESULT — 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 LDU is at least N. Modified.

💖 Author

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

Definition at line 155 of file dsyt22.f.

👑 Author

Generated automatically by Doxygen for LAPACK from the source code.

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okay that's the whole page, bestie 🫶 — it's LAPACK, Version 3.12.0, from the page file /home/abui...IG/dsyt22.f(3). mwah 💋