DGBSVX(3)

📦 version 3.12.0, in opensuse-leap 16.0

🌸 NAME 🌸

Okay so this is DGBSVX(3) from the Library Functions Manual, and the source file is /home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/dgbsvx.f — bestie, we are in the LAPACK source directory and it's giving band-matrix solver energy 💖

✨ SYNOPSIS ✨

🧚 Functions/Subroutines 🧚

  • 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

💅 Function/Subroutine Documentation 💅

   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. Periodt. 👑

🎀 Purpose 🎀

Real talk: 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. So yes, it's not just solving — it's also doing the little extra check-in, and we love that for us 🫶

🦋 Description 🦋

Here's the exact step-by-step tea this subroutine performs, bestie 🍓:

  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, warning: 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. So it's warning you, not ghosting you. 💀
  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. It's giving self-improvement era. ✨
  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. It 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. It 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 is DOUBLE PRECISION 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 DGBTRF. 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 DGBTRF; 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. It 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 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 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', 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 is DOUBLE PRECISION 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 — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N).
  • X — this is where part 1 stops; the description of X continues in part 2, bestie 💖
  • X — X is a DOUBLE PRECISION array, dimension (LDX,NRHS). If INFO = 0 or INFO = N+1, it's 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'. Yes bestie, that's a whole lot of conditionals — it's giving equilibration main character energy 💅.
  • LDX — LDX is INTEGER. It's the leading dimension of the array X. LDX >= max(1,N). No exceptions, periodt.
  • 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 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. Cute little safety cushion, lowkey 🫶.
  • 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). That's the "how much would I have to nudge things" vibe fr fr.
  • 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 is used. 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. Heads up: this is the vibe-check field for your factorization 😭.
  • 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: 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 366 of file dgbsvx.f.

✨ Author ✨

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