TESTING/LIN/dpbt05.f(3)

okay bestie, this is the TESTING/LIN/dpbt05.f(3) page from the Library Functions Manual, and it's all about the DPBT05 subroutine. We're keeping every detail, every constraint, every formula — no cap. 💅✨

🌸 NAME 🌸

TESTING/LIN/dpbt05.f — that's the file/page name, periodt. The function living here is dpbt05 / DPBT05, and yes, it's giving main character energy. 👑

✨ SYNOPSIS ✨

Here's the synopsis, exactly as the manual lists it:

   Functions/Subroutines
       subroutine dpbt05 (uplo, n, kd, nrhs, ab, ldab, b, ldb, x, ldx, xact,
           ldxact, ferr, berr, reslts)
           DPBT05

So yeah: it's a Functions/Subroutines situation, and the subroutine is dpbt05 with the arguments (uplo, n, kd, nrhs, ab, ldab, b, ldb, x, ldx, xact, ldxact, ferr, berr, reslts), and its name line is DPBT05. iykyk 💖

🦋 Function/Subroutine Documentation 🦋

✨ Full signature ✨

   subroutine dpbt05 (character uplo, integer n, integer kd, integer nrhs,
       double precision, dimension( ldab, * ) ab, integer ldab, double
       precision, dimension( ldb, * ) b, integer ldb, double precision,
       dimension( ldx, * ) x, integer ldx, double precision, dimension( ldxact,
       * ) xact, integer ldxact, double precision, dimension( * ) ferr, double
       precision, dimension( * ) berr, double precision, dimension( * ) reslts)

That's the whole glow-up signature, character for character. obsessed. 💖

💅 DPBT05

🎯 Purpose

okay so DPBT05 tests the error bounds from iterative refinement for the computed solution to a system of equations A*X = B, where A is a symmetric band matrix. Iconic testing behavior, honestly.

• RESLTS(1) = test of the error bound = norm(X - XACT) / ( norm(X) * FERR )

If this ratio is not less than one, a large value is returned — that's the signal, bestie. vibe check failed 😭

• 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

🧚 Parameters

Here's every parameter, with all the types, dimensions, and constraints kept exactly as they should be:

• UPLO — UPLO is CHARACTER*1. It specifies whether the upper or lower triangular part of the symmetric matrix A is stored. = 'U': Upper triangular; = 'L': Lower triangular.
• N — N is INTEGER. The number of rows of the matrices X, B, and XACT, and the order of the matrix A. N >= 0.
• KD — KD is INTEGER. The number of super-diagonals of the matrix A if UPLO = 'U', or the number of sub-diagonals if UPLO = 'L'. KD >= 0.
• NRHS — NRHS is INTEGER. The number of columns of the matrices X, B, and XACT. NRHS >= 0.
• AB — AB is DOUBLE PRECISION array, dimension (LDAB,N). The upper or lower triangle of the symmetric 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 as follows: 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).
• LDAB — LDAB is INTEGER. The leading dimension of the array AB. LDAB >= KD+1.
• B — B is DOUBLE PRECISION array, dimension (LDB,NRHS). The right hand side vectors for the system of linear equations.
• LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N).
• X — X is DOUBLE PRECISION array, dimension (LDX,NRHS). The computed solution vectors. Each vector is stored as a column of the matrix X.
• LDX — LDX is INTEGER. The leading dimension of the array X. LDX >= max(1,N).
• XACT — XACT is DOUBLE PRECISION array, dimension (LDX,NRHS). The exact solution vectors. Each vector is stored as a column of the matrix XACT.
• LDXACT — LDXACT is INTEGER. The leading dimension of the array XACT. LDXACT >= max(1,N).
• FERR — FERR is DOUBLE PRECISION 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 — BERR is DOUBLE PRECISION 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 — RESLTS is DOUBLE PRECISION 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 + (*) ).

👑 Author

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

Definition: this icon is defined at line 169 of file dpbt05.f.

🌈 Author 🌈

Generated automatically by Doxygen for LAPACK from the source code. So yes, Doxygen did the auto-gen thing. We stan a generated page. 🫶

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mwah 💋 — source/version sign-off: LAPACK Version 3.12.0, page TESTING/LIN/dpbt05.f(3). That's the footer vibe, and we're obsessed. ✨🫶