TESTING/EIG/ssyt21.f(3)

🦋 TESTING/EIG/ssyt21.f(3) 🦋

✨ NAME ✨

TESTING/EIG/ssyt21.f — okay so this page is from the Library Functions Manual, and it's all about SSYT21, no cap. 💖

✨ SYNOPSIS ✨

Functions/Subroutines

subroutine ssyt21 (itype, uplo, n, kband, a, lda, d, e, u, ldu, v, ldv,
    tau, work, result)

SSYT21

🌸 Function/Subroutine Documentation 🌸

subroutine ssyt21 (integer itype, character uplo, integer n, integer kband,
    real, dimension( lda, * ) a, integer lda, real, dimension( * ) d, real,
    dimension( * ) e, real, dimension( ldu, * ) u, integer ldu, real,
    dimension( ldv, * ) v, integer ldv, real, dimension( * ) tau, real,
    dimension( * ) work, real, dimension( 2 ) result)

SSYT21

🎀 Purpose 🎀

okay so SSYT21 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 . — yes, bestie, those blanks are literally in the source, I'm not playing 🙈. And We shall use the letter to refer to the product of Householder transformations (which should be equal to U). (again, blanks are in the original, no cap 😭).

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). ✨

💅 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, SSYT21 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 REAL 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 REAL array, dimension (N). The diagonal of the (symmetric tri-) diagonal matrix.
• E — E is REAL 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 REAL 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 REAL 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.
• LDV — LDV is INTEGER. The leading dimension of V. LDV must be at least N and at least 1.
• TAU — TAU is REAL 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 REAL array, dimension (2*N**2).
• RESULT — RESULT is REAL 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 at line 205 of file ssyt21.f.

🌈 Author 🌈

Generated automatically by Doxygen for LAPACK from the source code.

---

and that's the whole vibe, bestie! this page is from LAPACK Version 3.12.0, file TESTING/EIG/ssyt21.f(3), and it was generated automatically by Doxygen for LAPACK from the source code. slay 💖✨