gbsvx(3)

📦
version 3.12.1 · 9 versions
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💖 gbsvx(3) — Library Functions Manual 💖

okay bestie, welcome to the expert factor-and-solve zone 🫶 this is gbsvx, and she is literally here to factor and solve, expert. no cap.

🌸 NAME 🌸

gbsvx — gbsvx: factor and solve, expert 💅

✨ SYNOPSIS ✨

🧚 Functions 🧚

bestie, there are four flavors of this icon — one for each real/complex vibe. here they are, exactly as they come:

       subroutine cgbsvx (fact, trans, n, kl, ku, nrhs, ab, ldab, afb, ldafb,
           ipiv, equed, r, c, b, ldb, x, ldx, rcond, ferr, berr, work, rwork,
           info)

CGBSVX computes the solution to system of linear equations A * X = B for GB matrices ✨

       subroutine dgbsvx (fact, trans, n, kl, ku, nrhs, ab, ldab, afb, ldafb,
           ipiv, equed, r, c, b, ldb, x, ldx, rcond, ferr, berr, work, iwork,
           info)

DGBSVX computes the solution to system of linear equations A * X = B for GB matrices 🎀

       subroutine sgbsvx (fact, trans, n, kl, ku, nrhs, ab, ldab, afb, ldafb,
           ipiv, equed, r, c, b, ldb, x, ldx, rcond, ferr, berr, work, iwork,
           info)

SGBSVX computes the solution to system of linear equations A * X = B for GB matrices 🌸

       subroutine zgbsvx (fact, trans, n, kl, ku, nrhs, ab, ldab, afb, ldafb,
           ipiv, equed, r, c, b, ldb, x, ldx, rcond, ferr, berr, work, rwork,
           info)

ZGBSVX computes the solution to system of linear equations A * X = B for GB matrices 🦋

🌈 Detailed Description 🌈

📚 Function Documentation 📚

💅 subroutine cgbsvx

       subroutine cgbsvx (character fact, character trans, integer n, integer kl,
           integer ku, integer nrhs, complex, dimension( ldab, * ) ab, integer
           ldab, complex, dimension( ldafb, * ) afb, integer ldafb, integer,
           dimension( * ) ipiv, character equed, real, dimension( * ) r, real,
           dimension( * ) c, 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)

CGBSVX computes the solution to system of linear equations A * X = B for GB matrices 👑

Purpose:

CGBSVX uses the LU factorization to compute the solution to a complex system of linear equations A * X = B, A**T * X = B, or A**H * X = B, where A is a band matrix of order N with KL subdiagonals and KU superdiagonals, 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 by this subroutine, and yes, we are going step by step because she is thorough:

  1. If FACT = 'E', real scaling factors are computed to equilibrate the system:
TRANS = 'N':  diag(R)*A*diag(C)     *inv(diag(C))*X = diag(R)*B
TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*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(R)*A*diag(C) and B by diag(R)*B (if TRANS='N') or diag(C)*B (if TRANS = 'T' or 'C').

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

where L is a product of permutation and unit lower triangular matrices with KL subdiagonals, and U is upper triangular with KL+KU superdiagonals.

  1. If some U(i,i)=0, so that U is exactly singular, 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. ⚠️
  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(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') 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 and IPIV contain the factored form of A. If EQUED is not 'N', the matrix A has been equilibrated with scaling factors given by R and C. AB, AFB, and IPIV are not 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.
  • TRANS — TRANS is CHARACTER*1. Specifies the form of the system of equations.
    • = 'N': A * X = B (No transpose)
    • = 'T': A**T * X = B (Transpose)
    • = 'C': A**H * X = B (Conjugate transpose)
  • N — N is INTEGER. The number of linear equations, i.e., the order of the matrix A. N >= 0.
  • KL — KL is INTEGER. The number of subdiagonals within the band of A. KL >= 0.
  • KU — KU is INTEGER. The number of superdiagonals within the band of A. KU >= 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 array, dimension (LDAB,N).
    • On entry, the matrix A in band storage, in rows 1 to KL+KU+1. The j-th column of A is stored in the j-th column of the array AB as follows:
AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)
  • If FACT = 'F' and EQUED is not 'N', then A must have been equilibrated by the scaling factors in R and/or C. AB is not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit.
  • On exit, if EQUED .ne. 'N', A is scaled as follows:
EQUED = 'R':  A := diag(R) * A
EQUED = 'C':  A := A * diag(C)
EQUED = 'B':  A := diag(R) * A * diag(C).
  • LDAB — LDAB is INTEGER. The leading dimension of the array AB. LDAB >= KL+KU+1.
  • AFB — AFB is COMPLEX array, dimension (LDAFB,N).
    • If FACT = 'F', then AFB is an input argument and on entry contains details of the LU factorization of the band matrix A, as computed by CGBTRF. U is stored as an upper triangular band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and the multipliers used during the factorization are stored in rows KL+KU+2 to 2*KL+KU+1. If EQUED .ne. 'N', then AFB is the factored form of the equilibrated matrix A.
    • If FACT = 'N', then AFB is an output argument and on exit returns details of the LU factorization of A.
    • If FACT = 'E', then AFB is an output argument and on exit returns details of the LU factorization of the equilibrated matrix A (see the description of AB for the form of the equilibrated matrix).
  • LDAFB — LDAFB is INTEGER. The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1.
  • IPIV — IPIV is INTEGER array, dimension (N).
    • If FACT = 'F', then IPIV is an input argument and on entry contains the pivot indices from the factorization A = L*U as computed by CGBTRF; row i of the matrix was interchanged with row IPIV(i).
    • If FACT = 'N', then IPIV is an output argument and on exit contains the pivot indices from the factorization A = L*U of the original matrix A.
    • If FACT = 'E', then IPIV is an output argument and on exit contains the pivot indices from the factorization A = L*U of the equilibrated matrix A.
  • EQUED — EQUED is CHARACTER*1. Specifies the form of equilibration that was done.
    • = 'N': No equilibration (always true if FACT = 'N').
    • = 'R': Row equilibration, i.e., A has been premultiplied by diag(R).
    • = 'C': Column equilibration, i.e., A has been postmultiplied by diag(C).
    • = 'B': Both row and column equilibration, i.e., A has been replaced by diag(R) * A * diag(C).
    • EQUED is an input argument if FACT = 'F'; otherwise, it is an output argument.
  • R — R is REAL array, dimension (N). The row scale factors for A. If EQUED = 'R' or 'B', A is multiplied on the left by diag(R); if EQUED = 'N' or 'C', R is not accessed. R is an input argument if FACT = 'F'; otherwise, R is an output argument. If FACT = 'F' and EQUED = 'R' or 'B', each element of R must be positive.
  • C — C is REAL array, dimension (N). The column scale factors for A. If EQUED = 'C' or 'B', A is multiplied on the right by diag(C); if EQUED = 'N' or 'R', C is not accessed. C is an input argument if FACT = 'F'; otherwise, C is an output argument. If FACT = 'F' and EQUED = 'C' or 'B', each element of C must be positive.
  • B
  • B — okay bestie, B is a COMPLEX array, dimension (LDB,NRHS). On entry, it's the right hand side matrix B. On exit: if EQUED = 'N', B is not modified; if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by diag(R)*B; if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is overwritten by diag(C)*B. 🫶
  • LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N).
  • X — X is 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 A and B are modified on exit if EQUED .ne. 'N', and the solution to the equilibrated system is inv(diag(C))*X if TRANS = 'N' and EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C' and EQUED = 'R' or 'B'. 💖
  • LDX — LDX is INTEGER. The leading dimension of the array X. LDX >= max(1,N).
  • RCOND — RCOND is 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 — FERR is 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 — 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). 🦋
  • WORK — WORK is COMPLEX array, dimension (2*N).
  • RWORK — RWORK is REAL array, dimension (MAX(1,N)). On exit, RWORK(1) contains the reciprocal pivot growth factor norm(A)/norm(U). The norm is used. If RWORK(1) is much less than 1, then the stability of the LU factorization of the (equilibrated) matrix A could be poor. This also means that the solution X, condition estimator RCOND, and forward error bound FERR could be unreliable. If factorization fails with 0<INFO<=N, then RWORK(1) contains the reciprocal pivot growth factor for the leading INFO columns of A. 🌸
  • 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: U(i,i) is exactly zero. The factorization has been completed, but the factor U is exactly singular, so the solution and error bounds could not be 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.

Definition at line 367 of file cgbsvx.f.

okay bestie, next up is the dgbsvx era:

   subroutine dgbsvx (character fact, character trans, integer n, integer kl,
       integer ku, integer nrhs, double precision, dimension( ldab, * ) ab,
       integer ldab, double precision, dimension( ldafb, * ) afb, integer
       ldafb, integer, dimension( * ) ipiv, character equed, double precision,
       dimension( * ) r, double precision, dimension( * ) c, 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)

DGBSVX computes the solution to system of linear equations A * X = B for GB matrices. 👑

🌸 Purpose

DGBSVX uses the LU factorization to compute the solution to a real system of linear equations A * X = B, A**T * X = B, or A**H * X = B, where A is a band matrix of order N with KL subdiagonals and KU superdiagonals, 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 by this subroutine:

  1. If FACT = 'E', real scaling factors are computed to equilibrate the system:
TRANS = 'N':  diag(R)*A*diag(C)     *inv(diag(C))*X = diag(R)*B
TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*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(R)*A*diag(C) and B by diag(R)*B (if TRANS='N') or diag(C)*B (if TRANS = 'T' or 'C'). 🎀

  1. If FACT = 'N' or 'E', the LU decomposition is used to factor the matrix A (after equilibration if FACT = 'E') as A = L * U, where L is a product of permutation and unit lower triangular matrices with KL subdiagonals, and U is upper triangular with KL+KU superdiagonals. 🦋
  2. If some U(i,i)=0, so that U is exactly singular, 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. 😭
  3. The system of equations is solved for X using the factored form of A.
  4. Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it. 💖
  5. If equilibration was used, the matrix X is premultiplied by diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') 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 and IPIV contain the factored form of A. If EQUED is not 'N', the matrix A has been equilibrated with scaling factors given by R and C. AB, AFB, and IPIV are not 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.
  • TRANS — TRANS is CHARACTER*1. Specifies the form of the system of equations.
    • = 'N': A * X = B (No transpose)
    • = 'T': A**T * X = B (Transpose)
    • = 'C': A**H * X = B (Transpose)
  • N — N is INTEGER. The number of linear equations, i.e., the order of the matrix A. N >= 0.
  • KL — KL is INTEGER. The number of subdiagonals within the band of A. KL >= 0.
  • KU — KU is INTEGER. The number of superdiagonals within the band of A. KU >= 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 matrix A in band storage, in rows 1 to KL+KU+1. The j-th column of A is stored in the j-th column of the array AB as follows:
AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)

If FACT = 'F' and EQUED is not 'N', then A must have been equilibrated by the scaling factors in R and/or C. AB is not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit.

On exit, if EQUED .ne. 'N', A is scaled as follows:

EQUED = 'R':  A := diag(R) * A
EQUED = 'C':  A := A * diag(C)
EQUED = 'B':  A := diag(R) * A * diag(C).
  • LDAB — LDAB is INTEGER. The leading dimension of the array AB. LDAB >= KL+KU+1.
  • AFB

AFB

Ok so AFB is DOUBLE PRECISION array, dimension (LDAFB,N) — and what it means depends entirely on your FACT vibe:

  • If FACT = 'F', then AFB is an input argument, and on entry it holds all the deets of the LU factorization of the band matrix A, exactly as DGBTRF computed it. U lives in rows 1 to KL+KU+1 as an upper triangular band matrix with KL+KU superdiagonals, and the multipliers from the factorization are chilling in rows KL+KU+2 to 2*KL+KU+1. And if EQUED .ne. 'N', then AFB is the factored form of the equilibrated matrix A. iykyk 💅
  • If FACT = 'N', then AFB is an output argument, and on exit it hands you back the details of the LU factorization of A.
  • If FACT = 'E', then AFB is an output argument, and on exit it gives you the details of the LU factorization of the equilibrated matrix A (peep the description of AB for what that equilibrated matrix actually looks like).

LDAFB

LDAFB is INTEGER — it's the leading dimension of the array AFB, and the rule is LDAFB >= 2*KL+KU+1. Respect it. 🫶

IPIV

IPIV is INTEGER array, dimension (N) — the pivot index diary:

  • If FACT = 'F', IPIV is an input argument and on entry it contains the pivot indices from the factorization A = L*U as computed by DGBTRF; row i of the matrix was swapped with row IPIV(i). 🔁
  • If FACT = 'N', IPIV is an output argument and on exit it contains the pivot indices from the factorization A = L*U of the original matrix A.
  • If FACT = 'E', IPIV is an output argument and on exit it contains the pivot indices from the factorization A = L*U of the equilibrated matrix A.

EQUED

EQUED is CHARACTER*1 and it specifies which flavour of equilibration was done:

  • = 'N': no equilibration at all (and this is always the case if FACT = 'N')
  • = 'R': row equilibration, meaning A got premultiplied by diag(R)
  • = 'C': column equilibration, meaning A got postmultiplied by diag(C)
  • = 'B': both, meaning A was replaced by diag(R) * A * diag(C) ✨

So EQUED is an input argument if FACT = 'F'; otherwise it's an output argument. Periodt.

R

R is DOUBLE PRECISION array, dimension (N) — the row scale factors for A, bestie. If EQUED = 'R' or 'B', then A is multiplied on the left by diag(R); if EQUED = 'N' or 'C', then R is not accessed at all (she's off duty 💀). R is an input argument if FACT = 'F'; otherwise R is an output argument. And heads up: if FACT = 'F' and EQUED = 'R' or 'B', then every single element of R must be positive. No zeros, no negatives, none of that. 🚫

C

C is DOUBLE PRECISION array, dimension (N) — the column scale factors for A. If EQUED = 'C' or 'B', A is multiplied on the right by diag(C); if EQUED = 'N' or 'R', then C is not accessed. C is an input argument if FACT = 'F'; otherwise C is an output argument. Same deal as R: if FACT = 'F' and EQUED = 'C' or 'B', each element of C must be positive. 💖

B

B is DOUBLE PRECISION array, dimension (LDB,NRHS)

On entry: the right hand side matrix B. On exit it's a whole situation:

  • if EQUED = 'N', then B is not modified
  • if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by diag(R)*B
  • if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is overwritten by diag(C)*B

Like, your B may not come back the same. She's been through something. 😭

LDB

LDB is INTEGER — the leading dimension of the array B, with LDB >= max(1,N). Non-negotiable.

X

X is DOUBLE PRECISION array, dimension (LDX,NRHS)

If INFO = 0 or INFO = N+1, this is the N-by-NRHS solution matrix X to the original system of equations. ⭐ But note — and this is the part everyone forgets — A and B are modified on exit if EQUED .ne. 'N', and the solution to the equilibrated system is inv(diag(C))*X if TRANS = 'N' and EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C' and EQUED = 'R' or 'B'. So like… do the un-scaling. Please. 🙏

LDX

LDX is INTEGER — leading dimension of the array X, and LDX >= max(1,N).

RCOND

RCOND is DOUBLE PRECISION — the estimate of the reciprocal condition number of the matrix A after equilibration (if equilibration happened). If RCOND is less than machine precision (and especially if RCOND = 0), then the matrix is singular to working precision, and that whole situation gets flagged by a return code of INFO > 0. So watch her. 👀

FERR

FERR is DOUBLE PRECISION array, dimension (NRHS) — the estimated forward error bound for each solution vector X(j) (aka 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). And honestly? The estimate is only as reliable as the estimate for RCOND, and it's almost always a slight overestimate of the true error. She's dramatic like that. 💅

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. Tiny number = good vibes. 🔥

WORK

WORK is DOUBLE PRECISION array, dimension (MAX(1,3*N))

On exit, WORK(1) contains the reciprocal pivot growth factor norm(A)/norm(U). The norm in question is (the source doc literally says "The norm is used" with a suspicious little gap where the name should be, so just know: it's that ratio, bestie). If WORK(1) is much less than 1, then the stability of the LU factorization of the (equilibrated) matrix A could be poor. And that also means the solution X, the condition estimator RCOND, and the forward error bound FERR could all be unreliable. 💀 If the factorization fails with 0<INFO<=N, then WORK(1) contains the reciprocal pivot growth factor for the leading INFO columns of A.

IWORK

IWORK is INTEGER array, dimension (N). Just vibes and integer workspace. 🎀

INFO

INFO is INTEGER — the moment of truth:

  • = 0: successful exit ✨
  • < 0: if INFO = -i, the i-th argument had an illegal value 🚨
  • > 0: if INFO = i, and i is:
    • <= N: U(i,i) is exactly zero. The factorization has been completed, but the factor U is exactly singular, so the solution and error bounds could not be computed. RCOND = 0 is returned. Rough. 😭
    • = N+1: U is nonsingular, but RCOND is less than machine precision, meaning 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. Main character energy from the solution, honestly. 👑

🫶 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

📍 Definition at line 366 of file dgbsvx.f.

🌈 subroutine sgbsvx

subroutine sgbsvx (character fact, character trans, integer n, integer kl,
    integer ku, integer nrhs, real, dimension( ldab, * ) ab, integer ldab,
    real, dimension( ldafb, * ) afb, integer ldafb, integer, dimension( * )
    ipiv, character equed, real, dimension( * ) r, real, dimension( * ) c,
    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)

SGBSVX computes the solution to system of linear equations A * X = B for GB matrices. 🦋 (Yes, this is the single-precision sibling of the routine above — same energy, REAL instead of DOUBLE PRECISION.)

🎯 Purpose

!>
!> SGBSVX uses the LU factorization to compute the solution to a real
!> system of linear equations A * X = B, A**T * X = B, or A**H * X = B,
!> where A is a band matrix of order N with KL subdiagonals and KU
!> superdiagonals, and X and B are N-by-NRHS matrices.
!>
!> Error bounds on the solution and a condition estimate are also
!> provided.
!>

📖 Description

Okay so here's the breakdown, bestie — 🧚

okay bestie, so here’s the actual sequence this subroutine runs through — no cap, every step matters:

  1. If FACT = 'E', real scaling factors are computed to equilibrate the system:
TRANS = 'N':  diag(R)*A*diag(C)     *inv(diag(C))*X = diag(R)*B
TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*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(R)*A*diag(C) and B by diag(R)*B (if TRANS='N') or diag(C)*B (if TRANS = 'T' or 'C').

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

where L is a product of permutation and unit lower triangular matrices with KL subdiagonals, and U is upper triangular with KL+KU superdiagonals.

  1. If some U(i,i)=0, so that U is exactly singular, 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.
  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(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so that it solves the original system before equilibration.

🎀 Parameters 🎀

okay so here’s the cast of characters — every single one, no skips:

  • FACT — CHARACTER*1. This one decides whether you’re handing over the already-factored matrix or making the routine do the work, and if it does the work, whether it should equilibrate first. The values:
    • = 'F': On entry, AFB and IPIV contain the factored form of A. If EQUED is not 'N', the matrix A has been equilibrated with scaling factors given by R and C. AB, AFB, and IPIV are not 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.
  • TRANS — CHARACTER*1. Specifies the form of the system of equations:
    • = 'N': A * X = B (No transpose)
    • = 'T': A**T * X = B (Transpose)
    • = 'C': A**H * X = B (Transpose)
  • N — INTEGER. The number of linear equations, i.e., the order of the matrix A. N >= 0.
  • KL — INTEGER. The number of subdiagonals within the band of A. KL >= 0.
  • KU — INTEGER. The number of superdiagonals within the band of A. KU >= 0.
  • NRHS — INTEGER. The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0.
  • AB — REAL array, dimension (LDAB,N). On entry, the matrix A in band storage, in rows 1 to KL+KU+1. The j-th column of A is stored in the j-th column of the array AB as follows:
AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)

If FACT = 'F' and EQUED is not 'N', then A must have been equilibrated by the scaling factors in R and/or C. AB is not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit.

On exit, if EQUED .ne. 'N', A is scaled as follows:

EQUED = 'R':  A := diag(R) * A
EQUED = 'C':  A := A * diag(C)
EQUED = 'B':  A := diag(R) * A * diag(C).
  • LDAB — INTEGER. The leading dimension of the array AB. LDAB >= KL+KU+1.
  • AFB — REAL array, dimension (LDAFB,N). If FACT = 'F', then AFB is an input argument and on entry contains details of the LU factorization of the band matrix A, as computed by SGBTRF. U is stored as an upper triangular band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and the multipliers used during the factorization are stored in rows KL+KU+2 to 2*KL+KU+1. If EQUED .ne. 'N', then AFB is the factored form of the equilibrated matrix A.

If FACT = 'N', then AFB is an output argument and on exit returns details of the LU factorization of A.

If FACT = 'E', then AFB is an output argument and on exit returns details of the LU factorization of the equilibrated matrix A (see the description of AB for the form of the equilibrated matrix).

  • LDAFB — INTEGER. The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1.
  • IPIV — INTEGER array, dimension (N). If FACT = 'F', then IPIV is an input argument and on entry contains the pivot indices from the factorization A = L*U as computed by SGBTRF; row i of the matrix was interchanged with row IPIV(i).

If FACT = 'N', then IPIV is an output argument and on exit contains the pivot indices from the factorization A = L*U of the original matrix A.

If FACT = 'E', then IPIV is an output argument and on exit contains the pivot indices from the factorization A = L*U of the equilibrated matrix A.

  • EQUED — CHARACTER*1. Specifies the form of equilibration that was done:
    • = 'N': No equilibration (always true if FACT = 'N').
    • = 'R': Row equilibration, i.e., A has been premultiplied by diag(R).
    • = 'C': Column equilibration, i.e., A has been postmultiplied by diag(C).
    • = 'B': Both row and column equilibration, i.e., A has been replaced by diag(R) * A * diag(C).

EQUED is an input argument if FACT = 'F'; otherwise, it is an output argument.

  • R — REAL array, dimension (N). The row scale factors for A. If EQUED = 'R' or 'B', A is multiplied on the left by diag(R); if EQUED = 'N' or 'C', R is not accessed. R is an input argument if FACT = 'F'; otherwise, R is an output argument. If FACT = 'F' and EQUED = 'R' or 'B', each element of R must be positive.
  • C — REAL array, dimension (N). The column scale factors for A. If EQUED = 'C' or 'B', A is multiplied on the right by diag(C); if EQUED = 'N' or 'R', C is not accessed. C is an input argument if FACT = 'F'; otherwise, C is an output argument. If FACT = 'F' and EQUED = 'C' or 'B', each element of C must be positive.
  • B — REAL array, dimension (LDB,NRHS). On entry, the right hand side matrix B. On exit:
    • if EQUED = 'N', B is not modified;
    • if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by diag(R)*B;
    • if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is overwritten by diag(C)*B.
  • LDB — INTEGER. The leading dimension of the array B. LDB >= max(1,N).
  • X — REAL 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 A and B are modified on exit if EQUED .ne. 'N', and the solution to the equilibrated system is inv(diag(C))*X if TRANS = 'N' and EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C' and EQUED = 'R' or 'B'.
  • 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 — and that’s where this part of the page cuts off, so WORK’s description is a mystery for now 🫶

WORK 💖

Okay so WORK is a REAL array with dimension (MAX(1,3*N)).

On exit, WORK(1) contains the reciprocal pivot growth factor norm(A)/norm(U). The norm it's referring to is the one the source calls The norm is used. (yes, there's a literal double-space gap there, bestie 🫠). If WORK(1) is much less than 1, then the stability of the LU factorization of the (equilibrated) matrix A could be poor. This also means that the solution X, condition estimator RCOND, and forward error bound FERR could be unreliable. If factorization fails with 0<INFO<=N, then WORK(1) contains the reciprocal pivot growth factor for the leading INFO columns of A.

IWORK

IWORK is an INTEGER array, dimension (N). That's it, that's the whole slay. ✨

INFO

INFO is INTEGER, and it's the vibe check at the end:

  • = 0: successful exit
  • < 0: if INFO = -i, the i-th argument had an illegal value
  • > 0: if INFO = i, and i is
    • <= N: U(i,i) is exactly zero. The factorization has been completed, but the factor U is exactly singular, so the solution and error bounds could not be 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 (and yeah, the source sentence cuts off right there 🫠)

Author 🫶

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 365 of file sgbsvx.f.

subroutine zgbsvx ✨

subroutine zgbsvx (character fact, character trans, integer n, integer kl,
     integer ku, integer nrhs, complex*16, dimension( ldab, * ) ab, integer
     ldab, complex*16, dimension( ldafb, * ) afb, integer ldafb, integer,
     dimension( * ) ipiv, character equed, double precision, dimension( * )
     r, double precision, dimension( * ) c, 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)

ZGBSVX computes the solution to system of linear equations A * X = B for GB matrices

Purpose:

ZGBSVX uses the LU factorization to compute the solution to a complex system of linear equations A * X = B, A**T * X = B, or A**H * X = B, where A is a band matrix of order N with KL subdiagonals and KU superdiagonals, and X and B are N-by-NRHS matrices.

Error bounds on the solution and a condition estimate are also provided. No cap.

Description:

The following steps are performed by this subroutine:

  1. If FACT = 'E', real scaling factors are computed to equilibrate the system:
TRANS = 'N':  diag(R)*A*diag(C)     *inv(diag(C))*X = diag(R)*B
TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*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(R)*A*diag(C) and B by diag(R)*B (if TRANS='N') or diag(C)*B (if TRANS = 'T' or 'C').

  1. If FACT = 'N' or 'E', the LU decomposition is used to factor the matrix A (after equilibration if FACT = 'E') as A = L * U, where L is a product of permutation and unit lower triangular matrices with KL subdiagonals, and U is upper triangular with KL+KU superdiagonals.
  2. If some U(i,i)=0, so that U is exactly singular, 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.
  3. The system of equations is solved for X using the factored form of A.
  4. Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it.
  5. If equilibration was used, the matrix X is premultiplied by diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') 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 and IPIV contain the factored form of A. If EQUED is not 'N', the matrix A has been equilibrated with scaling factors given by R and C. AB, AFB, and IPIV are not 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.
  • TRANS — CHARACTER*1. Specifies the form of the system of equations.
    • = 'N': A * X = B (No transpose)
    • = 'T': A**T * X = B (Transpose)
    • = 'C': A**H * X = B (Conjugate transpose)
  • N — INTEGER. The number of linear equations, i.e., the order of the matrix A. N >= 0.
  • KL — INTEGER. The number of subdiagonals within the band of A. KL >= 0.
  • KU — INTEGER. The number of superdiagonals within the band of A. KU >= 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*16 array, dimension (LDAB,N). On entry, the matrix A in band storage, in rows 1 to KL+KU+1. The j-th column of A is stored in the j-th column of the array AB as follows: AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)
    • If FACT = 'F' and EQUED is not 'N', then A must have been equilibrated by the scaling factors in R and/or C. AB is not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit.
    • On exit, if EQUED .ne. 'N', A is scaled as follows:
      • EQUED = 'R': A := diag(R) * A
      • EQUED = 'C': A := A * diag(C)
      • EQUED = 'B': A := diag(R) * A * diag(C).
  • LDAB — INTEGER. The leading dimension of the array AB. LDAB >= KL+KU+1.
  • AFB — COMPLEX*16 array, dimension (LDAFB,N).
    • If FACT = 'F', then AFB is an input argument and on entry contains details of the LU factorization of the band matrix A, as computed by ZGBTRF. U is stored as an upper triangular band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and the multipliers used during the factorization are stored in rows KL+KU+2 to 2*KL+KU+1. If EQUED .ne. 'N', then AFB is the factored form of the equilibrated matrix A.
    • If FACT = 'N', then AFB is an output argument and on exit returns details of the LU factorization of A.
    • If FACT = 'E', then AFB is an output argument and on exit returns details of the LU factorization of the equilibrated matrix A (see the description of AB for the form of the equilibrated matrix).
  • LDAFB — INTEGER. The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1.
  • IPIV — INTEGER array, dimension (N).
    • If FACT = 'F', then IPIV is an input argument and on entry contains the pivot indices from the factorization A = L*U as computed by ZGBTRF; row i of the matrix was interchanged with row IPIV(i).
    • If FACT = 'N', then IPIV is an output argument and on exit contains the pivot indices from the factorization A = L*U of the original matrix A.
    • If FACT = 'E', then IPIV is an output argument and on exit contains the pivot indices from the factorization A = L*U of the equilibrated matrix A.
  • EQUED — CHARACTER*1. Specifies the form of equilibration that was done.
    • = 'N': No equilibration (always true if FACT = 'N').
    • = 'R': Row equilibration, i.e., A has been premultiplied by diag(R).
    • = 'C': Column equilibration, i.e., A has been postmultiplied by diag(C).
    • = 'B': Both row and column equilibration, i.e., A has been replaced by diag(R) * A * diag(C).
    • EQUED is an input argument if FACT = 'F'; otherwise, it is an output argument.
  • R — DOUBLE PRECISION array, dimension (N). The row scale factors for A. If EQUED = 'R' or 'B', A is multiplied on the left by diag(R); if EQUED = 'N' or 'C', R is not accessed. R is an input argument if FACT = 'F'; otherwise, R is an output argument. If FACT = 'F' and EQUED = 'R' or 'B', each element of R must be positive.

C

  • C — okay so C is a DOUBLE PRECISION array, dimension (N), and it’s the column scale factors for A, bestie. If EQUED = 'C' or 'B', A is multiplied on the right by diag(C); if EQUED = 'N' or 'R', C is not accessed. C is an input argument if FACT = 'F'; otherwise, C is an output argument. And if FACT = 'F' and EQUED = 'C' or 'B', each element of C must be positive — no zero/negative nonsense, periodt.
  • B — B is a COMPLEX*16 array, dimension (LDB,NRHS). On entry, it’s the right hand side matrix B. On exit: if EQUED = 'N', B is not modified; if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by diag(R)*B; if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is overwritten by diag(C)*B.
  • LDB — INTEGER, the leading dimension of the array B. LDB >= max(1,N). That’s the vibe check.
  • X — COMPLEX*16 array, dimension (LDX,NRHS). If INFO = 0 or INFO = N+1, this is the N-by-NRHS solution matrix X to the original system of equations. Note that A and B are modified on exit if EQUED .ne. 'N', and the solution to the equilibrated system is inv(diag(C))*X if TRANS = 'N' and EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C' and EQUED = 'R' or 'B'.
  • LDX — INTEGER, the leading dimension of the array X. LDX >= max(1,N). Slay.
  • RCOND — 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 — 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), 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. Ngl, it’s giving cautious bestie energy.
  • BERR — 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 — COMPLEX*16 array, dimension (2*N). That’s it. That’s the whole note. 💅
  • RWORK — DOUBLE PRECISION array, dimension (MAX(1,N)). On exit, RWORK(1) contains the reciprocal pivot growth factor norm(A)/norm(U). The norm is used. If RWORK(1) is much less than 1, then the stability of the LU factorization of the (equilibrated) matrix A could be poor. This also means that the solution X, condition estimator RCOND, and forward error bound FERR could be unreliable. If factorization fails with 0<INFO<=N, then RWORK(1) contains the reciprocal pivot growth factor for the leading INFO columns of A.
  • INFO — INTEGER, the status tea:
    • = 0: successful exit.
    • < 0: if INFO = -i, the i-th argument had an illegal value.
    • > 0: if INFO = i, and i is:
      • <= N: U(i,i) is exactly zero. The factorization has been completed, but the factor U is exactly singular, so the solution and error bounds could not be 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.

Definition at line 367 of file zgbsvx.f.

👑 Author (the Doxygen one) 👑

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Okay bestie, that’s the whole parameter party 🎀💖. Footer tea: source/version is LAPACK Version 3.12.0, and the page is gbsvx(3) — no separate date line in the footer, just that iconic ending energy. Byeee ✨🦋

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