DSPT21(3)

Library Functions Manual — okay bestie, this is the page for /home/abui...IG/dspt21.f(3). Yes, the header is extra and repeats that path, because man pages love a little echo. The manual name is Library Functions Manual, and we're about to get into the DSPT21 of it all. 💅✨

🌸 NAME 🌸

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/TESTING/EIG/dspt21.f

🦋 SYNOPSIS 🦋

✨ Functions/Subroutines ✨

subroutine DSPT21 (itype, uplo, n, kband, ap, d, e, u, ldu, vp, tau,
    work, result)
    DSPT21

💖 Function/Subroutine Documentation 💖

🌟 DSPT21 🌟

subroutine DSPT21 (integer itype, character uplo, integer n, integer kband,
    double precision, dimension( * ) ap, double precision, dimension( * ) d,
    double precision, dimension( * ) e, double precision, dimension( ldu, *
    ) u, integer ldu, double precision, dimension( * ) vp, double precision,
    dimension( * ) tau, double precision, dimension( * ) work, double
    precision, dimension( 2 ) result)

DSPT21

🎀 Purpose 🎀

DSPT21 generally checks a decomposition of the form

A = U S U**T

where **T means transpose, A is symmetric (stored in packed format), 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 the U is expressed as a product of Householder transformations, whose vectors are stored in the array [blank in source] and whose scaling constants are in [blank in source]; we shall use the letter [blank in source] to refer to the product of Householder transformations (which should be equal to U). No cap, those blanks are literally blank in the source — I'm not out here inventing names. 💀

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 )

Packed storage means that, for example, if UPLO='U', then the columns of the upper triangle of A are stored one after another, so that A(1,j+1) immediately follows A(j,j) in the array AP. Similarly, if UPLO='L', then the columns of the lower triangle of A are stored one after another in AP, so that A(j+1,j+1) immediately follows A(n,j) in the array AP. This means that A(i,j) is stored in:

AP( i + j*(j-1)/2 )                 if UPLO='U'

AP( i + (2*n-j)*(j-1)/2 )           if UPLO='L'

The array VP bears the same relation to the matrix V that A does to AP.

For ITYPE > 1, the transformation U is expressed as a product of Householder transformations:

If UPLO='U', then V = H(n-1)...H(1), where

H(j) = I  -  tau(j) v(j) v(j)**T

and the first j-1 elements of v(j) are stored in V(1:j-1,j+1), (i.e., VP( j*(j+1)/2 + 1 : j*(j+1)/2 + j-1 )), the j-th element is 1, and the last n-j elements are 0.

If UPLO='L', then V = H(1)...H(n-1), where

H(j) = I  -  tau(j) v(j) v(j)**T

and the first j elements of v(j) are 0, the (j+1)-st is 1, and the (j+2)-nd through n-th elements are stored in V(j+2:n,j) (i.e., in VP( (2*n-j)*(j-1)/2 + j+2 : (2*n-j)*(j-1)/2 + n ) .)

🌈 Parameters 🌈

• 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 )
  • 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 (CHARACTER) — if UPLO='U', AP and VP are considered to contain the upper triangle of A and V. If UPLO='L', AP and VP are considered to contain the lower triangle of A and V.
• N (INTEGER) — the size of the matrix. If it is zero, DSPT21 does nothing. It must be at least zero. Zero is a valid slay, apparently. 💅
• 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.
• AP (DOUBLE PRECISION array, dimension (N*(N+1)/2)) — the original (unfactored) matrix. It is assumed to be symmetric, and contains the columns of just the upper triangle (UPLO='U') or only the lower triangle (UPLO='L'), packed one after another.
• D (DOUBLE PRECISION array, dimension (N)) — the diagonal of the (symmetric tri-) diagonal matrix.
• E (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 (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 (INTEGER) — the leading dimension of U. LDU must be at least N and at least 1.
• VP (DOUBLE PRECISION array, dimension (N*(N+1)/2)) — if ITYPE=2 or 3, the columns of this array contain the Householder vectors used to describe the orthogonal matrix in the decomposition, as described in purpose.
  • 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.
• 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.
• WORK (DOUBLE PRECISION array, dimension (N**2+N)) — workspace.
• 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 ITYPE=1.

🦋 Author 🦋

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

Definition at line 219 of file dspt21.f.

🫶 Author 🫶

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Catch you later, bestie — the footer says LAPACK Version 3.12.0, with date/source tag /home/abui...IG/dspt21.f(3). Stay packed, stay iconic. 💖💅