SRC/zpbrfs.f(3)

💅 SRC/zpbrfs.f — aka ZPBRFS, the manual page bestie 💅

Okay so bestie, this is the page for SRC/zpbrfs.f, straight outta the Library Functions Manual. Buckle up, she's a whole vibe. 🎀✨

---

✨ SYNOPSIS ✨

Functions/Subroutines

subroutine zpbrfs (uplo, n, kd, nrhs, ab, ldab, afb, ldafb, b, ldb, x,
    ldx, ferr, berr, work, rwork, info)

ZPBRFS — that's literally her name, no notes. 💖

---

🦋 Function/Subroutine Documentation 🦋

subroutine zpbrfs (character uplo, integer n, integer kd, integer nrhs,
    complex*16, dimension( ldab, * ) ab, integer ldab, complex*16,
    dimension( ldafb, * ) afb, integer ldafb, complex*16, dimension( ldb, *
    ) b, integer ldb, complex*16, dimension( ldx, * ) x, integer ldx, double
    precision, dimension( * ) ferr, double precision, dimension( * ) berr,
    complex*16, dimension( * ) work, double precision, dimension( * ) rwork,
    integer info)

ZPBRFS ✨

🌸 Purpose 🌸

So like, the whole point of ZPBRFS? She improves the computed solution to a system of linear equations when the coefficient matrix is Hermitian positive definite and banded. And bestie, she doesn't just stop there — she also provides error bounds and backward error estimates for the solution. Multitasking queen. 👑

🎀 Parameters 🎀

• UPLO — UPLO is CHARACTER*1. The tea:
  • = 'U': Upper triangle of A is stored
  • = 'L': Lower triangle of A is stored
• N — N is INTEGER. The order of the matrix A. Condition check: N >= 0.
• KD — KD is INTEGER. The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. Condition check: KD >= 0.
• NRHS — NRHS is INTEGER. The number of right hand sides, i.e., the number of columns of the matrices B and X. Condition check: NRHS >= 0.
• AB — AB is COMPLEX*16 array, dimension (LDAB,N). 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 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. Condition check: LDAB >= KD+1.
• AFB — AFB is COMPLEX*16 array, dimension (LDAFB,N). The triangular factor U or L from the Cholesky factorization A = U**H*U or A = L*L**H of the band matrix A as computed by ZPBTRF, in the same storage format as A (see AB). She's literally wearing the same outfit as AB, iykyk. 🫶
• LDAFB — LDAFB is INTEGER. The leading dimension of the array AFB. Condition check: LDAFB >= KD+1.
• B — B is COMPLEX*16 array, dimension (LDB,NRHS). The right hand side matrix B. Main character energy, obviously.
• LDB — LDB is INTEGER. The leading dimension of the array B. Condition check: LDB >= max(1,N).
• X — X is COMPLEX*16 array, dimension (LDX,NRHS). On entry, the solution matrix X, as computed by ZPBTRS. On exit, the improved solution matrix X. Glow-up moment fr fr. 💅
• LDX — LDX is INTEGER. The leading dimension of the array X. Condition check: LDX >= max(1,N).
• FERR — FERR is DOUBLE PRECISION array, dimension (NRHS). The estimated forward error bound for each solution vector X(j) (the j-th column of the solution matrix X). If XTRUE is the true solution corresponding to X(j), FERR(j) is an estimated upper bound for the magnitude of the largest element in (X(j) - XTRUE) divided by the magnitude of the largest element in X(j). The estimate is as reliable as the estimate for RCOND, and is almost always a slight overestimate of the true error. So like, she's trustworthy but she rounds up a lil. 😭
• BERR — BERR is DOUBLE PRECISION array, dimension (NRHS). The componentwise relative backward error of each solution vector X(j) (i.e., the smallest relative change in any element of A or B that makes X(j) an exact solution).
• WORK — WORK is COMPLEX*16 array, dimension (2*N). Just doing her little job behind the scenes.
• RWORK — RWORK is DOUBLE PRECISION array, dimension (N). Same energy, different array.
• INFO — INFO is INTEGER. The receipts:
  • = 0: successful exit
  • < 0: if INFO = -i, the i-th argument had an illegal value

🔥 Internal Parameters 🔥

ITMAX is the maximum number of steps of iterative refinement. Periodt. 💀

👑 Author 👑

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

Definition at line 187 of file zpbrfs.f. Noted, screenshotting. 📸

---

🌈 Author 🌈

Generated automatically by Doxygen for LAPACK from the source code. So like, no human sat there writing this by hand — the robots did it. Respect tho. 🤖💖

---

Okay bestie, that's the whole page! 🌸 This gem comes to you courtesy of LAPACK Version 3.12.0 — go forth and improve those Hermitian positive definite banded solutions, you absolute icon. 💜🦋🍓