TESTING/EIG/dspt21.f(3)

🌸 NAME 🌸

okay so bestie, welcome to TESTING/EIG/dspt21.f(3) from the Library Functions Manual — manual name folded in near the top, because we’re not copying that boring header line. The NAME is TESTING/EIG/dspt21.f, and it is the star of this page. No cap. This file is lowkey the main character and we’re obsessed. 💖

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🦋 SYNOPSIS 🦋

Functions/Subroutines

the synopsis is literally this, and we’re keeping it exact — no cap:

       subroutine dspt21 (itype, uplo, n, kband, ap, d, e, u, ldu, vp, tau,
           work, result)
           DSPT21

DSPT21 said hi in the synopsis too, periodt. 🫶

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💅 Function/Subroutine Documentation 💅

here’s the full typed signature, exact and iconic:

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

and yes, DSPT21 is the subroutine’s whole name and it’s serving 💅. vibe check: passed ✨

🎀 Purpose 🎀

okay so 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 U is expressed as a product of Householder transformations, whose vectors are stored in the array VP and whose scaling constants are in TAU — we shall use the letter V to refer to the product of Householder transformations (which should be equal to U). 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 )

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 ) .)

this is a lot, but we keep every formula, bestie 😭💖 fr fr

🍓 Parameters 🍓

all the deets, no skipping, 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 ).
  • 2: U expressed as a product V of Householder 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 Householder 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.
• 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.

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👑 Author 👑

Generated automatically by Doxygen for LAPACK from the source code.

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byeee bestie 💖 this cutie is from LAPACK, Version 3.12.0, and it’s the manual page for TESTING/EIG/dspt21.f(3). periodt. 🫶