CPBT05(3)

🦋 CPBT05(3) — like, the Hermitian band-matrix error-bound checker, bestie

okay so real quick: this is a page from the Library Functions Manual, and the file it's documenting literally lives at /home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/TESTING/LIN/cpbt05.f. that's the whole NAME section, no notes 💅

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✨ SYNOPSIS ✨

🎀 Functions/Subroutines

here's the whole cast, no cap:

subroutine CPBT05 (uplo, n, kd, nrhs, ab, ldab, b, ldb, x, ldx, xact,
    ldxact, ferr, berr, reslts)
        CPBT05

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💖 Function/Subroutine Documentation 💖

🌸 the actual signature, character for character

subroutine CPBT05 (character uplo, integer n, integer kd, integer nrhs,
    complex, dimension( ldab, * ) ab, integer ldab, complex, dimension( ldb,
    * ) b, integer ldb, complex, dimension( ldx, * ) x, integer ldx,
    complex, dimension( ldxact, * ) xact, integer ldxact, real, dimension( *
    ) ferr, real, dimension( * ) berr, real, dimension( * ) reslts)

CPBT05 — that's her name, that's her whole thing 🫶

🔥 Purpose:

so what does she actually do? she tests the error bounds from iterative refinement for a computed solution to a system of equations A*X = B, where A is a Hermitian band matrix. that's it, that's the vibe ✨

!>
!> CPBT05 tests the error bounds from iterative refinement for the
!> computed solution to a system of equations A*X = B, where A is a
!> Hermitian band matrix.
!>
!> RESLTS(1) = test of the error bound
!>           = norm(X - XACT) / ( norm(X) * FERR )
!>
!> A large value is returned if this ratio is not less than one.
!>
!> RESLTS(2) = residual from the iterative refinement routine
!>           = the maximum of BERR / ( NZ*EPS + (*) ), where
!>             (*) = NZ*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
!>             and NZ = max. number of nonzeros in any row of A, plus 1
!>

💀 note the "a large value is returned if this ratio is not less than one" — that's your red flag, babes. big number = the bound did not hold.

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🌈 Parameters 🌈

every single one of them, obsessed with the thoroughness honestly:

• UPLO — is CHARACTER*1, and she specifies whether the upper or lower triangular part of the Hermitian matrix A is stored. the options:
  • = 'U': Upper triangular
  • = 'L': Lower triangular
• N — INTEGER. the number of rows of the matrices X, B, and XACT, and the order of the matrix A. constraint: N >= 0.
• KD — INTEGER. the number of super-diagonals of the matrix A if UPLO = 'U', or the number of sub-diagonals if UPLO = 'L'. constraint: KD >= 0.
• NRHS — INTEGER. the number of columns of the matrices X, B, and XACT. constraint: NRHS >= 0.
• AB — COMPLEX array, dimension (LDAB,N). it's the upper or lower triangle of the Hermitian band matrix A, stored in the first KD+1 rows of the array. the j-th column of A is stored in the j-th column of the array AB, like so:

if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

fr fr, copy that indexing exactly, do not freestyle 🔥

• LDAB — INTEGER. the leading dimension of the array AB. constraint: LDAB >= KD+1.
• B — COMPLEX array, dimension (LDB,NRHS). the right hand side vectors for the system of linear equations. the whole point, honestly.
• LDB — INTEGER. the leading dimension of the array B. constraint: LDB >= max(1,N).
• X — COMPLEX array, dimension (LDX,NRHS). the computed solution vectors. each vector is stored as a column of the matrix X.
• LDX — INTEGER. the leading dimension of the array X. constraint: LDX >= max(1,N).
• XACT — COMPLEX array, dimension (LDX,NRHS). the exact solution vectors. each vector is stored as a column of the matrix XACT. (yes, the dimension comment says LDX not LDXACT — that's what the original says and we do not touch it 💅)
• LDXACT — INTEGER. the leading dimension of the array XACT. constraint: LDXACT >= max(1,N).
• FERR — REAL array, dimension (NRHS). the estimated forward error bounds for each solution vector X. if XTRUE is the true solution, FERR bounds the magnitude of the largest entry in (X - XTRUE) divided by the magnitude of the largest entry in X.
• BERR — REAL array, dimension (NRHS). the componentwise relative backward error of each solution vector (i.e., the smallest relative change in any entry of A or B that makes X an exact solution).
• RESLTS — REAL array, dimension (2). the maximum over the NRHS solution vectors of the ratios:

RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR )
RESLTS(2) = BERR / ( NZ*EPS + (*) )

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👑 Author

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

Definition at line 169 of file cpbt05.f.

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⭐ Author (the meta one)

Generated automatically by Doxygen for LAPACK from the source code. not a human, just a robot doing its little job 🤖✨

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okay that's the whole page, you ate that 🍓 and for the cute sign-off: this gorgeous thing is LAPACK, Version 3.12.0, and the source file stamped on it is /home/abui...IN/cpbt05.f(3) — mwah, periodt 💜🎀