pbsvx(3)

📦

version 3.12.1 · 9 versions

3.12.1 · arch rolling (latest)
3.12.0 · opensuse-leap 16.0
3.12.1 · ubuntu stonking
3.12.1 · ubuntu questing
3.12.0 · ubuntu noble
3.12.0 · ubuntu noble
3.12.0 · centos-stream 10, fedora 43, fedora 44
3.12.1 · ubuntu resolute
3.12.1 · debian trixie

okay so this is pbsvx(3) from the Library Functions Manual — yes, the same pbsvx(3) up in the header — and she is here to factor and solve like an expert. 💅✨

✨ NAME ✨

pbsvx — pbsvx: factor and solve, expert

🌈 SYNOPSIS 🌈

🫶 Functions 🫶

cpbsvx — CPBSVX computes the solution to system of linear equations A * X = B for OTHER matrices

       subroutine cpbsvx (fact, uplo, n, kd, nrhs, ab, ldab, afb, ldafb, equed,
           s, b, ldb, x, ldx, rcond, ferr, berr, work, rwork, info)

dpbsvx — DPBSVX computes the solution to system of linear equations A * X = B for OTHER matrices

       subroutine dpbsvx (fact, uplo, n, kd, nrhs, ab, ldab, afb, ldafb, equed,
           s, b, ldb, x, ldx, rcond, ferr, berr, work, iwork, info)

spbsvx — SPBSVX computes the solution to system of linear equations A * X = B for OTHER matrices

       subroutine spbsvx (fact, uplo, n, kd, nrhs, ab, ldab, afb, ldafb, equed,
           s, b, ldb, x, ldx, rcond, ferr, berr, work, iwork, info)

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

       subroutine zpbsvx (fact, uplo, n, kd, nrhs, ab, ldab, afb, ldafb, equed,
           s, b, ldb, x, ldx, rcond, ferr, berr, work, rwork, info)

🦋 Detailed Description 🦋

No extra detail here in this part, bestie — the page just moves straight into the function documentation. 💅

💅 Function Documentation 💅

🧮 `cpbsvx` — complex expert mode

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

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

🎯 Purpose

okay so CPBSVX 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. 💖

🦋 Description

The following steps are performed:

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.

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.

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. 👀 warning behavior, bestie — not a full stop.
The system of equations is solved for X using the factored form of A.
Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it.
If equilibration was used, the matrix X is premultiplied by diag(S) so that it solves the original system before equilibration.

💖 Parameters

FACT — 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 — CHARACTER*1.
- = 'U': Upper triangle of A is stored;
- = 'L': Lower triangle of A is stored.
N — INTEGER. The number of linear equations, i.e., the order of the matrix A. N >= 0.
KD — INTEGER. The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0.
NRHS — INTEGER. The number of right-hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0.
AB — COMPLEX 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 — INTEGER. The leading dimension of the array A. LDAB >= KD+1.
AFB — COMPLEX 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 — INTEGER. The leading dimension of the array AFB. LDAFB >= KD+1.
EQUED — 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 — REAL 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 — COMPLEX 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 — INTEGER. The leading dimension of the array B. LDB >= max(1,N).
X — COMPLEX 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 — INTEGER. The leading dimension of the array X. LDX >= max(1,N).
RCOND — REAL. 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.
FERR — REAL 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.
BERR — REAL 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 — COMPLEX array, dimension (2*N).
RWORK — REAL array, dimension (N).
INFO — (the original page cuts off right here mid-parameter, bestie — INFO's description continues in the next part 💅)

INFO is INTEGER — okay bestie, here's the return-code tea you actually need:

= 0: successful exit
< 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

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 cpbsvx.f.

🧚 `subroutine dpbsvx` 🧚

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

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

⭐ Purpose

DPBSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to compute the solution to a real system of linear equations A * X = B, where A is an N-by-N symmetric 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. It's giving “solve it and also hand you the receipts” energy, fr fr.

💫 Description

The following steps are performed:

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.
If FACT = 'N' or 'E', the Cholesky decomposition is used to factor the matrix A (after equilibration if FACT = 'E') as A = U**T * U, if UPLO = 'U', or A = L * L**T, if UPLO = 'L', where U is an upper triangular band matrix, and L is a lower triangular band matrix.
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.
The system of equations is solved for X using the factored form of A.
Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it.
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 DOUBLE PRECISION array, dimension (LDAB,N). On entry, the upper or lower triangle of the symmetric 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 DOUBLE PRECISION 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**T*U or A = L*L**T 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**T*U or A = L*L**T.
- 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**T*U or A = L*L**T 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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.

FERR

FERR — FERR is DOUBLE PRECISION array, dimension (NRHS). This is 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), then 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. 💅
BERR — BERR is DOUBLE PRECISION array, dimension (NRHS). This is 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 DOUBLE PRECISION array, dimension (3*N). 🎀
IWORK — IWORK is INTEGER array, dimension (N). 🦋
INFO — INFO is INTEGER.
- = 0: successful exit 🌸
- < 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 🦋

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

Two-dimensional storage of the symmetric 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 340 of file dpbsvx.f.

🌈 subroutine spbsvx 🌈

   subroutine spbsvx (character fact, character uplo, integer n, integer kd,
       integer nrhs, real, dimension( ldab, * ) ab, integer ldab, real,
       dimension( ldafb, * ) afb, integer ldafb, character equed, real,
       dimension( * ) s, real, dimension( ldb, * ) b, integer ldb, real,
       dimension( ldx, * ) x, integer ldx, real rcond, real, dimension( * )
       ferr, real, dimension( * ) berr, real, dimension( * ) work, integer,
       dimension( * ) iwork, integer info)

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

✨ Purpose ✨

Okay so SPBSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to compute the solution to a real system of linear equations

A * X = B,

where A is an N-by-N symmetric 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. Obsessed, honestly.

🌸 Description 🌸

The following steps are performed:

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.

If FACT = 'N' or 'E', the Cholesky decomposition is used to factor the matrix A (after equilibration if FACT = 'E') as

A = U**T * U,  if UPLO = 'U', or
A = L * L**T,  if UPLO = 'L',

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

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.
The system of equations is solved for X using the factored form of A.
Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it.
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 REAL array, dimension (LDAB,N). On entry, the upper or lower triangle of the symmetric 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 REAL 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**T*U or A = L*L**T 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**T*U or A = L*L**T.
- 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**T*U or A = L*L**T 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 REAL 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 REAL 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 — X is REAL array, dimension (LDX,NRHS) ✨ so like, if INFO = 0 or INFO = N+1, this is the N-by-NRHS solution matrix X to the original system of equations. Bestie, read this part twice: if EQUED = 'Y', then A and B are modified on exit, and the solution to the equilibrated system is inv(diag(S))*X. That's the whole vibe check right there 💅
LDX — LDX is INTEGER, the leading dimension of the array X. Constraint, no cap: LDX >= max(1,N).
RCOND — RCOND is REAL, the estimate of the reciprocal condition number of the matrix A after equilibration (if equilibration happened, that is). If RCOND is less than the machine precision — and especially if RCOND = 0 — the matrix is singular to working precision 😭 and that whole situation gets flagged by a return code of INFO > 0.
FERR — FERR is REAL array, dimension (NRHS), the estimated forward error bound for each solution vector X(j) (that's the j-th column of the solution matrix X). Okay so: if XTRUE is the true solution corresponding to X(j), then 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). This estimate is exactly as reliable as the estimate for RCOND, and it's almost always a slight overestimate of the true error. Slight. Overestimate. Every time. 🫶
BERR — BERR is REAL 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). Iconic little measurement honestly.
WORK — WORK is REAL array, dimension (3*N). That's it, that's the workspace 🎀
IWORK — IWORK is INTEGER array, dimension (N). Integer workspace, she's doing her thing.
INFO — INFO is INTEGER, and here's the breakdown bestie, memorize it:
- = 0: successful exit 🌈 slay
- < 0: if INFO = -i, the i-th argument had an illegal value 🙈 (you passed something bad, not my fault)
- > 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. That's a wrap, no solution for you 💀
  - = 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. Not me being generous with the answers ✨

👑 Author 👑

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

🦋 Further Details 🦋

The band storage scheme is illustrated by the following example, when N = 6, KD = 2, and UPLO = 'U'. Get a load of this, it's lowkey gorgeous:

Two-dimensional storage of the symmetric 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. They're just there. Decor. Background characters 🎀

Definition at line 340 of file spbsvx.f.

💜 subroutine zpbsvx 💜

Like, real talk, this is the complex*16 glow-up of everything above. Same energy, double the precision, main character energy fr fr 🔥

   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 ✨

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. Yes, she does the extra credit too, no cap.

🍓 Description 🍓

The following steps are performed, in this exact order, bestie:

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. ⚠️ overwritten, as in, gone, as in, that's the trade.

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. 💅

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's not gonna leave you hanging just because of a little singularity scare 🫶
The system of equations is solved for X using the factored form of A.
Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it. Glow-up phase ✨
If equilibration was used, the matrix X is premultiplied by diag(S) so that it solves the original system before equilibration. Full circle, periodt.

🌈 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. Respect the input, bestie 🙏
- = '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, aka which half of the triangle you're serving:
- = '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. Constraint: 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'. Constraint: KD >= 0.
NRHS — NRHS is INTEGER, the number of right-hand sides, i.e., the number of columns of the matrices B and X. Constraint: NRHS >= 0.
AB — AB is COMPLEX*16 array, dimension (LDAB,N). Okay so on entry, it's 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). Yeah. Your input's gone. That's the whole deal.

LDAB — LDAB is INTEGER, the leading dimension of the array A. Constraint: LDAB >= KD+1.
AFB — AFB is COMPLEX*16 array, dimension (LDAFB,N). Depends entirely on FACT, so pay attention:
- 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. Constraint: LDAFB >= KD+1.
EQUED
EQUED — ok so this one’s a CHARACTER*1, and it specifies the form of equilibration that was done. vibe check:
- = '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. no cap, positivity is required.
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. so watch out, bestie — it might get rewritten.
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. iconic little transformation, honestly.
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. translating: tiny RCOND = red flag city.
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. it’s giving “close enough, and honest about it.”
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). scratch space bestie, don’t touch.
RWORK — RWORK is DOUBLE PRECISION array, dimension (N). 🌸
INFO — INFO is INTEGER, aka the final vibe report.
- = 0: successful exit.
- < 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. plot twist: it still tries, and sometimes it slays.

🌈 Further Details 🌈

okay so 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. leave them alone, they’re just vibing.

👑 Author 👑

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

absolute legends, fr fr.

Definition at line 339 of file zpbsvx.f.

👑 Author (Doxygen’s version) 👑

Generated automatically by Doxygen for LAPACK from the source code.

okay bestie, that’s the whole pbsvx(3) page — source: LAPACK, version 3.12.0, and the pbsvx(3) footer tag. you ate, no crumbs 💅✨🍓

📦 where this page comes from

arch rolling: lapack-doc 3.12.1-2 · custom
💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash-1017 on 2026-11-04: a language model's rewrite of the original, under its terms

pbsvx(3)

✨ NAME ✨

🌈 SYNOPSIS 🌈

🫶 Functions 🫶

🦋 Detailed Description 🦋

💅 Function Documentation 💅

🧮 cpbsvx — complex expert mode

🎯 Purpose

🦋 Description

💖 Parameters

👑 Author

🎀 Further Details

🧚 subroutine dpbsvx 🧚

⭐ Purpose

💫 Description

🌈 Parameters

🌟 Author 🌟

🦋 Further Details 🦋

🌈 subroutine spbsvx 🌈

✨ Purpose ✨

🌸 Description 🌸

💖 Parameters 💖

👑 Author 👑

🦋 Further Details 🦋

💜 subroutine zpbsvx 💜

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

✨ Purpose ✨

🍓 Description 🍓

🌈 Parameters 🌈

🌈 Further Details 🌈

👑 Author 👑

👑 Author (Doxygen’s version) 👑

🧮 `cpbsvx` — complex expert mode

🧚 `subroutine dpbsvx` 🧚