SRC/cgbsvx.f(3)

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🦋 SRC/cgbsvx.f(3) — Library Functions Manual 🦋

✨ NAME ✨

SRC/cgbsvx.f

🌸 SYNOPSIS 🌸

🧚 Functions/Subroutines 🧚

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

💜 Function/Subroutine Documentation 💜

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 🎀

okay so 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. slay. ✨

🍓 Description 🍓

Bestie, the following steps are performed by this subroutine. periodt 💅

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'). 🫶

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. 🦋

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. this part matters, fr fr 😭
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(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 is COMPLEX 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` 🦋

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 — INTEGER, bestie, and it's the leading dimension of the array X. The vibe check: LDX >= max(1,N). That's it, that's the rule 💅

🩺 RCOND

RCOND — REAL. This is the estimate of the reciprocal condition number of the matrix A after equilibration (if equilibration got done, that is). So here's the tea: if RCOND is less than the machine precision — and especially if RCOND = 0 — then the matrix is singular to working precision 😭. And how do you know that happened? That condition is indicated by a return code of INFO > 0. Iconic of it to warn you, honestly.

🎯 FERR

FERR — 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). 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). And ngl? This estimate is as reliable as the estimate for RCOND, and it's almost always a slight overestimate of the true error. So it's giving cautious bestie energy — better safe than sorry 🫶

📉 BERR

BERR — REAL array, dimension (NRHS). The componentwise relative backward error of each solution vector X(j) — as in, the smallest relative change in any element of A or B that makes X(j) an exact solution. Tiny number, big meaning ✨

🧰 WORK

WORK — COMPLEX array, dimension (2*N). That's the whole description. It's a scratchpad. It works. Moving on 💀

🌸 RWORK

RWORK — REAL array, dimension (MAX(1,N)). Okay so on exit, RWORK(1) contains the reciprocal pivot growth factor norm(A)/norm(U). The norm is used. (Yes, bestie, the page genuinely says "The norm is used" with the norm name missing — the source page leaves it blank there, so I'm not out here inventing one for you, iykyk 😭🙈) If RWORK(1) is much less than 1, then the stability of the LU factorization of the (equilibrated) matrix A could be poor. And that's not just a bad hair day — it also means that the solution X, condition estimator RCOND, and forward error bound FERR could be unreliable 👀. Also: if factorization fails with 0<INFO<=N, then RWORK(1) contains the reciprocal pivot growth factor for the leading INFO columns of A. Good to know, fr fr.

🚦 INFO

INFO — INTEGER. The receipts, in order:
- = 0: successful exit ✨ slay
- < 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. Not the ending we wanted 😭
  - = 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. Plot twist! The math said no and the results said bet 🔥👑

👑 Author

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

Definition at line 367 of file cgbsvx.f.

🧚 Author

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💖 and that's the whole page bestie 💖 Huge shoutout to the footer for the deets: this is LAPACK, Version 3.12.0, from SRC/cgbsvx.f(3). Go forth and solve your banded systems with confidence, mwah 🎀🦋💜

📦 where this page comes from

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