SRC/zgbsvxx.f(3)

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🎀 SRC/zgbsvxx.f — Library Functions Manual

🌸 NAME

SRC/zgbsvxx.f — yes bestie, that's literally the file we're talking about 💖

🦋 SYNOPSIS

🧚 Functions/Subroutines

  • subroutine zgbsvxx — (fact, trans, n, kl, ku, nrhs, ab, ldab, afb, ldafb, ipiv, equed, r, c, b, ldb, x, ldx, rcond, rpvgrw, berr, n_err_bnds, err_bnds_norm, err_bnds_comp, nparams, params, work, rwork, info) — ZGBSVXX computes the solution to system of linear equations A * X = B for GB matrices ✨

💜 Function/Subroutine Documentation

subroutine zgbsvxx (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 rpvgrw, double precision, dimension( *
    ) berr, integer n_err_bnds, double precision, dimension( nrhs, * )
    err_bnds_norm, double precision, dimension( nrhs, * ) err_bnds_comp,
    integer nparams, double precision, dimension( * ) params, complex*16,
    dimension( * ) work, double precision, dimension( * ) rwork, integer
    info)

ZGBSVXX computes the solution to system of linear equations A * X = B for GB matrices — that's the whole vibe of this icon 💅

🫶 Purpose

Okay so ZGBSVXX uses the LU factorization to compute the solution to a complex*16 system of linear equations A * X = B, where A is an N-by-N matrix and X and B are N-by-NRHS matrices. Math but make it fashion 💖

If requested, both normwise and maximum componentwise error bounds are returned. ZGBSVXX will return a solution with a tiny guaranteed error (O(eps) where eps is the working machine precision) unless the matrix is very ill-conditioned, in which case a warning is returned. Relevant condition numbers also are calculated and returned. so you get receipts, basically 🧾

ZGBSVXX accepts user-provided factorizations and equilibration factors; see the definitions of the FACT and EQUED options. Solving with refinement and using a factorization from a previous ZGBSVXX call will also produce a solution with either O(eps) errors or warnings, but we cannot make that claim for general user-provided factorizations and equilibration factors if they differ from what ZGBSVXX would itself produce. important caveat, bestie — don't @ me 🚨

🌈 Description

The following steps are performed:

  1. If FACT = 'E', double precision 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'). got it? slay 💅

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

where P is a permutation matrix, L is a unit lower triangular matrix, and U is upper triangular. iconic trio tbh 👑

  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 (see argument RCOND). If the reciprocal of the condition number is less than machine precision, the routine still goes on to solve for X and compute error bounds as described below. so it doesn't give up — it keeps going 🫶
  2. The system of equations is solved for X using the factored form of A. periodt ✨
  3. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero), the routine will use iterative refinement to try to get a small error and error bounds. Refinement calculates the residual to at least twice the working precision. fancy precision behavior, iykyk 🔥
  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. we un-equilibrate the answer, basically 🦋

Some optional parameters are bundled in the PARAMS array. These settings determine how refinement is performed, but often the defaults are acceptable. If the defaults are acceptable, users can pass NPARAMS = 0 which prevents the source code from accessing the PARAMS argument. defaults are giving low-effort but correct ✅

🎛️ 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, AF 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. A, AF, and IPIV are not modified. freeze — don't touch 🧊
  • = 'N': The matrix A will be copied to AF and factored. copy + factor era ✨
  • = 'E': The matrix A will be equilibrated if necessary, then copied to AF and factored. balance first, then factor 💅

🦋 TRANS

TRANS is CHARACTER*1.

Specifies the form of the system of equations:

  • = 'N': A * X = B (No transpose) no flip, no drama 🌸
  • = 'T': A**T * X = B (Transpose) flip it 🔄
  • = 'C': A**H * X = B (Conjugate Transpose = Transpose) conjugate flip, same energy 🪞

🌟 N

N is INTEGER.

The number of linear equations, i.e., the order of the matrix A. N >= 0. zero is valid, bestie 🫶

🎀 KL

KL is INTEGER.

The number of subdiagonals within the band of A. KL >= 0. under the main character diagonal 💅

🎀 KU

KU is INTEGER.

The number of superdiagonals within the band of A. KU >= 0. over the main character diagonal 👑

🧚 NRHS

NRHS is INTEGER.

The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0. how many B's we solving for 💖

🍓 AB

AB is 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 AB 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).

band storage math — yes it's giving index gymnastics 🤸

🦋 LDAB

LDAB is INTEGER.

The leading dimension of the array AB. LDAB >= KL+KU+1. respect the leading dimension 📏

💜 AFB

AFB is 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. input era — it already has the factorization tea ☕

If FACT = 'N', then AF is an output argument and on exit returns the factors L and U from the factorization A = P*L*U of the original matrix A. output era ✨

If FACT = 'E', then AF is an output argument and on exit returns the factors L and U from the factorization A = P*L*U of the equilibrated matrix A (see the description of A for the form of the equilibrated matrix). equilibrated output era 💅

🎀 LDAFB

LDAFB is INTEGER.

The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1. bigger leading dimension, bigger responsibility 🫡

🌈 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 = P*L*U as computed by ZGETRF; row i of the matrix was interchanged with row IPIV(i). pivot receipts 🧾

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

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

💖 EQUED

EQUED is CHARACTER*1.

Specifies the form of equilibration that was done.

  • = 'N': No equilibration (always true if FACT = 'N'). none, fresh out the box 🎁
  • = 'R': Row equilibration, i.e., A has been premultiplied by diag(R). rows got scaled 📏
  • = 'C': Column equilibration, i.e., A has been postmultiplied by diag(C). columns got scaled 📐
  • = 'B': Both row and column equilibration, i.e., A has been replaced by diag(R) * A * diag(C). both! equal opportunity scaling 💅

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

🍓 R

R 💖

R is DOUBLE PRECISION array, dimension (N) — the row scale factors for A, bestie. If EQUED = 'R' or 'B', then A gets multiplied on the left by diag(R); if EQUED = 'N' or 'C', R is literally not accessed at all. R is an input argument when FACT = 'F'; otherwise, R is an output argument. If FACT = 'F' and EQUED = 'R' or 'B', every element of R must be positive.

If R is output, each element of R is a power of the radix. If R is input, each element of R should be a power of the radix to ensure a reliable solution and error estimates. Scaling by powers of the radix does not cause rounding errors unless the result underflows or overflows.

Rounding errors during scaling lead to refining with a matrix that is not equivalent to the input matrix, producing error estimates that may not be reliable. That’s the tea, and it’s lowkey important 🫶

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.

If C is output, each element of C is a power of the radix. If C is input, each element of C should be a power of the radix to ensure a reliable solution and error estimates. Scaling by powers of the radix does not cause rounding errors unless the result underflows or overflows.

Rounding errors during scaling lead to refining with a matrix that is not equivalent to the input matrix, producing error estimates that may not be reliable. Same warning, different scale-factor baddie 😭

B 🦋

B is COMPLEX*16 array, dimension (LDB,NRHS)

On entry, this is the N-by-NRHS 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. And the rule is LDB >= max(1,N), no cap.

X 💅

X is COMPLEX*16 array, dimension (LDX,NRHS)

If INFO = 0, 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 🌸

LDX is INTEGER — the leading dimension of the array X. It’s giving LDX >= max(1,N), periodt.

RCOND 👑

RCOND is DOUBLE PRECISION — reciprocal scaled condition number. This is an estimate of the reciprocal Skeel condition number of the matrix A after equilibration (if done). If this is less than the machine precision (in particular, if it is zero), the matrix is singular to working precision. Note that the error may still be small even if this number is very small and the matrix appears ill-conditioned. Iconic little “don’t panic yet” clause.

RPVGRW 🔥

RPVGRW is DOUBLE PRECISION — reciprocal pivot growth.

On exit, this contains the reciprocal pivot growth factor norm(A)/norm(U). The norm is used. If this 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, estimated condition numbers, and error bounds could be unreliable.

If factorization fails with 0<INFO<=N, then this contains the reciprocal pivot growth factor for the leading INFO columns of A. In ZGESVX, this quantity is returned in RWORK(1).

BERR 🌈

BERR is DOUBLE PRECISION array, dimension (NRHS) — componentwise relative backward error. 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).

N_ERR_BNDS ⭐

N_ERR_BNDS is INTEGER — number of error bounds to return for each right hand side and each type (normwise or componentwise). See ERR_BNDS_NORM and ERR_BNDS_COMP below, bestie.

ERR_BNDS_NORM 💜

ERR_BNDS_NORM is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)

For each right-hand side, this array contains information about various error bounds and condition numbers corresponding to the normwise relative error, which is defined as follows:

Normwise relative error in the ith solution vector:

        max_j (abs(XTRUE(j,i) - X(j,i)))
       ------------------------------
             max_j abs(X(j,i))

The array is indexed by the type of error information as described below. There currently are up to three pieces of information returned.

The first index in ERR_BNDS_NORM(i,:) corresponds to the ith right-hand side.

The second index in ERR_BNDS_NORM(:,err) contains the following three fields:

  • err = 1 — boolean. Trust the answer if the reciprocal condition number is less than the threshold sqrt(n) * dlamch('Epsilon').
  • err = 2 — error bound: the estimated forward error, almost certainly within a factor of 10 of the true error so long as the next entry is greater than the threshold sqrt(n) * dlamch('Epsilon'). This error bound should only be trusted if the previous boolean is true.
  • err = 3 — reciprocal condition number: estimated normwise reciprocal condition number. Compared with the threshold sqrt(n) * dlamch('Epsilon') to determine if the error estimate is . These reciprocal condition numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some appropriately scaled matrix Z. Let Z = S*A, where S scales each row by a power of the radix so all absolute row sums of Z are approximately 1.

See Lapack Working Note 165 for further details and extra cautions.

ERR_BNDS_COMP 🧚

ERR_BNDS_COMP is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)

For each right-hand side, this array contains information about various error bounds and condition numbers corresponding to the componentwise relative error, which is defined as follows:

Componentwise relative error in the ith solution vector:

               abs(XTRUE(j,i) - X(j,i))
        max_j ----------------------
                    abs(X(j,i))

The array is indexed by the right-hand side i (on which the componentwise relative error depends), and the type of error information as described below. There currently are up to three pieces of information returned for each right-hand side. If componentwise accuracy is not requested (PARAMS(3) = 0.0), then ERR_BNDS_COMP is not accessed. If N_ERR_BNDS < 3, then at most the first (:,N_ERR_BNDS) entries are returned.

The first index in ERR_BNDS_COMP(i,:) corresponds to the ith right-hand side.

The second index in ERR_BNDS_COMP(:,err) contains the following three fields:

  • err = 1 — boolean. Trust the answer if the reciprocal condition number is less than the threshold sqrt(n) * dlamch('Epsilon').
  • err = 2 — error bound: the estimated forward error, almost certainly within a factor of 10 of the true error so long as the next entry is greater than the threshold sqrt(n) * dlamch('Epsilon'). This error bound should only be trusted if the previous boolean is true.
  • err = 3 — reciprocal condition number: estimated componentwise reciprocal condition number. Compared with the threshold sqrt(n) * dlamch('Epsilon') to determine if the error estimate is . These reciprocal condition numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some appropriately scaled matrix Z. Let Z = S*(A*diag(x)), where x is the solution for the current right-hand side and S scales each row of A*diag(x) by a power of the radix so all absolute row sums of Z are approximately 1.

See Lapack Working Note 165 for further details and extra cautions.

NPARAMS 🍓

NPARAMS is INTEGER — specifies the number of parameters set in PARAMS. If <= 0, the PARAMS array is never referenced and default values are used. Respect the defaults, bestie.

PARAMS 🎀

  • PARAMS — a DOUBLE PRECISION array, dimension NPARAMS. This is the ✨algorithm parameters✨ list, and honestly it's the main character of the tuning setup. If an entry is < 0.0, then that entry will be filled with the default value used for that parameter. Only positions up to NPARAMS are accessed — defaults are used for higher-numbered parameters. 🎛️
    • PARAMS(LA_LINRX_ITREF_I = 1) — whether to perform iterative refinement or not.
      • Default: 1.0D+0
      • = 0.0: no refinement is performed, and no error bounds are computed.
      • = 1.0: use the extra-precise refinement algorithm.
      • (other values are reserved for future use — so don't get creative, bestie 🙈)
    • PARAMS(LA_LINRX_ITHRESH_I = 2) — the maximum number of residual computations allowed for refinement.
      • Default: 10
      • Aggressive: set it to 100 to permit convergence using approximate factorizations or factorizations other than LU. ⚠️ But hear me out: if the factorization uses a technique other than Gaussian elimination, the guarantees in err_bnds_norm and err_bnds_comp may no longer be trustworthy. So like… proceed with intention.
    • PARAMS(LA_LINRX_CWISE_I = 3) — a flag determining if the code will attempt to find a solution with small componentwise relative error in the double-precision algorithm. Positive is true, 0.0 is false.
      • Default: 1.0 (attempt componentwise convergence)
  • WORK — COMPLEX*16 array, dimension (2*N). 🛠️
  • RWORK — DOUBLE PRECISION array, dimension (2*N). 🧮
  • INFO — INTEGER, the vibes-report at the end of the run:
    • = 0: successful exit. The solution to every right-hand side is guaranteed. 🎉
    • < 0: if INFO = -i, the i-th argument had an illegal value.
    • > 0 and <= N: U(INFO,INFO) 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+J: the solution corresponding to the Jth right-hand side is not guaranteed. The solutions corresponding to other right-hand sides K with K > J may not be guaranteed as well, but only the first such right-hand side is reported. If a small componentwise error is not requested (PARAMS(3) = 0.0) then the Jth right-hand side is the first with a normwise error bound that is not guaranteed (the smallest J such that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0) the Jth right-hand side is the first with either a normwise or componentwise error bound that is not guaranteed (the smallest J such that either ERR_BNDS_NORM(J,1) = 0.0 or ERR_BNDS_COMP(J,1) = 0.0). See the definition of ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information about all of the right-hand sides, check ERR_BNDS_NORM or ERR_BNDS_COMP. 🔍

🌸 Author 🌸

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 555 of file zgbsvxx.f. 📍

📄 Author

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