ZPBSVX(3)

📦 version 3.12.0, in opensuse-leap 16.0

✨ NAME ✨

okay so this page is from the Library Functions Manual, and the header path is literally /home/abui...RC/zpbsvx.f(3) — yes bestie, the path is the name, she's quirky like that 💅

The full NAME line is this exact path:

/home/abuild/rpmbuild/BUILD/lapack-
static-3.12.0-build/lapack-3.12.0/SRC/zpbsvx.f

That's it. That's the name. No cap. It's giving "my identity is my filesystem location" energy 😭

🌸 SYNOPSIS 🌸

Functions/Subroutines

  • subroutine ZPBSVX — ZPBSVX computes the solution to system of linear equations A * X = B for OTHER matrices. (yes, "OTHER matrices", that's literally what it says, bestie 💜)
subroutine ZPBSVX (fact, uplo, n, kd, nrhs, ab, ldab, afb, ldafb, equed,
           s, b, ldb, x, ldx, rcond, ferr, berr, work, rwork, info)

💖 Function/Subroutine Documentation 💖

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

ZPBSVX computes the solution to system of linear equations A * X = B for OTHER matrices.

🎯 Purpose 🎯

okay so ZPBSVX uses the Cholesky factorization A = U**H*U or A = L*L**H to compute the solution to a complex system of linear equations

A * X = B,

where A is an N-by-N Hermitian positive definite band matrix and X and B are N-by-NRHS matrices.

Error bounds on the solution and a condition estimate are also provided. she's thorough, we love that for her ✨

📝 Description 📝

The following steps are performed:

  1. If FACT = 'E', real scaling factors are computed to equilibrate the system:
diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B

Whether or not the system will be equilibrated depends on the scaling of the matrix A, but if equilibration is used, A is overwritten by diag(S)*A*diag(S) and B by diag(S)*B.

  1. If FACT = 'N' or 'E', the Cholesky decomposition is used to factor the matrix A (after equilibration if FACT = 'E') as
A = U**H * U,  if UPLO = 'U', or
A = L * L**H,  if UPLO = 'L',

where U is an upper triangular band matrix, and L is a lower triangular band matrix.

  1. If the leading principal minor of order i is not positive, then the routine returns with INFO = i. Otherwise, the factored form of A is used to estimate the condition number of the matrix A. If the reciprocal of the condition number is less than machine precision, INFO = N+1 is returned as a warning, but the routine still goes on to solve for X and compute error bounds as described below. she said "warning, but make it continue" — iconic behavior tbh
  2. The system of equations is solved for X using the factored form of A.
  3. Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it.
  4. If equilibration was used, the matrix X is premultiplied by diag(S) so that it solves the original system before equilibration.

🔧 Parameters 🔧

  • FACT — FACT is CHARACTER*1 Specifies whether or not the factored form of the matrix A is supplied on entry, and if not, whether the matrix A should be equilibrated before it is factored.
    • = 'F': On entry, AFB contains the factored form of A. If EQUED = 'Y', the matrix A has been equilibrated with scaling factors given by S. AB and AFB will not be modified.
    • = 'N': The matrix A will be copied to AFB and factored.
    • = 'E': The matrix A will be equilibrated if necessary, then copied to AFB and factored.
  • UPLO — UPLO is CHARACTER*1
    • = 'U': Upper triangle of A is stored;
    • = 'L': Lower triangle of A is stored.
  • N — N is INTEGER The number of linear equations, i.e., the order of the matrix A. 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'. KD >= 0.
  • NRHS — NRHS is INTEGER The number of right-hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0.
  • AB — AB is COMPLEX*16 array, dimension (LDAB,N) On entry, the upper or lower triangle of the Hermitian band matrix A, stored in the first KD+1 rows of the array, except if FACT = 'F' and EQUED = 'Y', then A must contain the equilibrated matrix diag(S)*A*diag(S). 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).

See below for further details.

On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by diag(S)*A*diag(S).

  • LDAB — LDAB is INTEGER The leading dimension of the array A. LDAB >= KD+1.
  • AFB — AFB is COMPLEX*16 array, dimension (LDAFB,N) If FACT = 'F', then AFB is an input argument and on entry contains the triangular factor U or L from the Cholesky factorization A = U**H *U or A = L*L**H of the band matrix A, in the same storage format as A (see AB). If EQUED = 'Y', then AFB is the factored form of the equilibrated matrix A.

If FACT = 'N', then AFB is an output argument and on exit returns the triangular factor U or L from the Cholesky factorization A = U**H *U or A = L*L**H.

If FACT = 'E', then AFB is an output argument and on exit returns the triangular factor U or L from the Cholesky factorization A = U**H *U or A = L*L**H of the equilibrated matrix A (see the description of A for the form of the equilibrated matrix).

  • LDAFB — LDAFB is INTEGER The leading dimension of the array AFB. LDAFB >= KD+1.
  • EQUED — EQUED is CHARACTER*1 Specifies the form of equilibration that was done.
    • = 'N': No equilibration (always true if FACT = 'N').
    • = 'Y': Equilibration was done, i.e., A has been replaced by diag(S) * A * diag(S). EQUED is an input argument if FACT = 'F'; otherwise, it is an output argument.
  • S — S is DOUBLE PRECISION array, dimension (N) The scale factors for A; not accessed if EQUED = 'N'. S is an input argument if FACT = 'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED = 'Y', each element of S must be positive.
  • B — B is COMPLEX*16 array, dimension (LDB,NRHS) On entry, the N-by-NRHS right hand side matrix B. On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y', B is overwritten by diag(S) * B.
  • LDB — LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).
  • X — X is COMPLEX*16 array, dimension (LDX,NRHS) If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to the original system of equations. Note that if EQUED = 'Y', A and B are modified on exit, and the solution to the equilibrated system is inv(diag(S))*X.
  • LDX — LDX is INTEGER The leading dimension of the array X. LDX >= max(1,N).
  • RCOND — RCOND is DOUBLE PRECISION The estimate of the reciprocal condition number of the matrix A after equilibration (if done). If RCOND is less than the machine precision (in particular, if RCOND = 0), the matrix is singular to working precision. This condition is indicated by a return code of INFO > 0. bestie, RCOND = 0 is a vibe check failure — singular to working precision, no cap 💀
  • 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. she's humble about being an overestimate, we stan 💜
  • 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)
  • RWORK — RWORK is DOUBLE PRECISION array, dimension (N)
  • INFO — (part 1 cuts off right here at the INFO heading, bestie — part 2 has the rest, no spoilers 💅)

INFO is INTEGER, and here’s the whole vibe check 💅:

  • = 0 — successful exit, we love a clean getaway ✨
  • < 0 — if INFO = -i, the i-th argument had an illegal value 🙈
  • > 0 — if INFO = i, and i is:
    • <= N — the leading principal minor of order i of A is not positive, so the factorization could not be completed, and the solution has not been computed. RCOND = 0 is returned. 💀
    • = N+1 — U is nonsingular, but RCOND is less than machine precision, meaning that the matrix is singular to working precision. Nevertheless, the solution and error bounds are computed because there are a number of situations where the computed solution can be more accurate than the value of RCOND would suggest. 🫶

Author 🫶

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

Further Details 🌈

Okay bestie, the band storage scheme is illustrated by the following example, when N = 6, KD = 2, and UPLO = 'U' 🦋:

Two-dimensional storage of the Hermitian matrix A:

     a11  a12  a13
          a22  a23  a24
               a33  a34  a35
                    a44  a45  a46
                         a55  a56
     (aij=conjg(aji))         a66

Band storage of the upper triangle of A:

      *    *   a13  a24  a35  a46
      *   a12  a23  a34  a45  a56
     a11  a22  a33  a44  a55  a66

Similarly, if UPLO = 'L' the format of A is as follows:

     a11  a22  a33  a44  a55  a66
     a21  a32  a43  a54  a65   *
     a31  a42  a53  a64   *    *

Array elements marked * are not used by the routine. 🙅‍♀️

Definition at line 339 of file zpbsvx.f.

Author 🎀

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