complexGBsolve(3)

📦
version 3.11.0 · 2 versions
, in debian bookworm

💖 complexGBsolve(3) 💖

okay bestie, welcome to complexGBsolve(3), a LAPACK joint, and it's giving complex solve driver functions for GB matrices. Three whole subroutines. Band matrices. LU factorization. We're doing this. 🎀

✨ NAME ✨

complexGBsolve — complex

🌸 SYNOPSIS 🌸

🧚 Functions

  • subroutine cgbsv (N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB, INFO) — CGBSV computes the solution to system of linear equations A * X = B for GB matrices (simple driver)
  • 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 cgbsvxx (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) — CGBSVXX computes the solution to system of linear equations A * X = B for GB matrices

👑 Detailed Description 👑

This is the group of complex solve driver functions for GB matrices. That's the whole vibe. 💅

💜 Function Documentation 💜

🦋 subroutine cgbsv (integer N, integer KL, integer KU, integer NRHS, complex, dimension( ldab, * ) AB, integer LDAB, integer, dimension( * ) IPIV, complex, dimension( ldb, * ) B, integer LDB, integer INFO) 🦋

CGBSV computes the solution to system of linear equations A * X = B for GB matrices (simple driver)

⭐ Purpose

CGBSV computes the solution to a complex system of linear equations

A * 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.

The LU decomposition with partial pivoting and row interchanges is used to factor A 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. The factored form of A is then used to solve the system of equations A * X = B. no cap 🫶

⭐ Parameters

  • 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 matrix B. NRHS >= 0.
  • AB — AB is COMPLEX array, dimension (LDAB,N). On entry, the matrix A in band storage, in rows KL+1 to 2*KL+KU+1; rows 1 to KL of the array need not be set. The j-th column of A is stored in the j-th column of the array AB as follows:
AB(KL+KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+KL)

On exit, details of the factorization: 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. See below for further details.

  • LDAB — LDAB is INTEGER. The leading dimension of the array AB. LDAB >= 2*KL+KU+1.
  • IPIV — IPIV is INTEGER array, dimension (N). The pivot indices that define the permutation matrix P; row i of the matrix was interchanged with row IPIV(i).
  • B — B is COMPLEX array, dimension (LDB,NRHS). On entry, the N-by-NRHS right hand side matrix B. On exit, if INFO = 0, the N-by-NRHS solution matrix X.
  • LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N).
  • INFO — INFO is INTEGER
    • = 0: successful exit ✨
    • < 0: if INFO = -i, the i-th argument had an illegal value
    • > 0: if INFO = i, U(i,i) is exactly zero. The factorization has been completed, but the factor U is exactly singular, and the solution has not been computed. 💀
bestie if you get INFO > 0 the solve did NOT happen. U is singular. the math said no. 😭

⭐ 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 M = N = 6, KL = 2, KU = 1:

On entry:                       On exit:

    *    *    *    +    +    +       *    *    *   u14  u25  u36
    *    *    +    +    +    +       *    *   u13  u24  u35  u46
    *   a12  a23  a34  a45  a56      *   u12  u23  u34  u45  u56
   a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55  u66
   a21  a32  a43  a54  a65   *      m21  m32  m43  m54  m65   *
   a31  a42  a53  a64   *    *      m31  m42  m53  m64   *    *

Array elements marked * are not used by the routine; elements marked + need not be set on entry, but are required by the routine to store elements of U because of fill-in resulting from the row interchanges. iykyk 🧚

🦋 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. obsessed. 🔥

⭐ 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.

  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 — COMPLEX array, dimension (LDAFB,N)

If FACT = 'F', then AFB is an input argument, and on entry it's holding the deets of the LU factorization of the band matrix A, exactly the way CGBTRF computed it. 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 chillin' 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 it returns details of the LU factorization of A.

If FACT = 'E', then AFB is an output argument, and on exit it returns details of the LU factorization of the equilibrated matrix A (go peek at the description of AB for what the equilibrated matrix looks like).

  • 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 it holds 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 it 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 it contains the pivot indices from the factorization A = L*U of the equilibrated matrix A.

  • EQUED — CHARACTER*1

Tells you which flavor of equilibration got done:

  • = 'N': no equilibration at all (this is 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's 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 at all, bestie, it's not even looked at. 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', every single 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. And same energy as R: if FACT = 'F' and EQUED = 'C' or 'B', each element of C must be positive. ✨

  • B — 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 — 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, this is the N-by-NRHS solution matrix X to the original system of equations. 📌 But keep this in mind: 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, don't forget to unscramble it. 😭

  • 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 that happened). If RCOND is less than the machine precision — and especially if RCOND = 0 — the matrix is singular to working precision, which is the math way of saying it's giving nothing. 🫠 This condition gets flagged 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), 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 as reliable as the estimate for RCOND, and it's almost always a slight overestimate of the true error. Lowkey generous of it.

  • 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. Slay.

  • WORK — COMPLEX array, dimension (2*N)
  • RWORK — REAL array, dimension (N)

On exit, RWORK(1) contains the reciprocal pivot growth factor norm(A)/norm(U). The 'max absolute element' norm is what's used here. If RWORK(1) is much less than 1, then the stability of the LU factorization of the (equilibrated) matrix A could be poor. 👀 That also means the solution X, the condition estimator RCOND, and the forward error bound FERR could all be unreliable. And if the factorization fails with 0<INFO<=N, then RWORK(1) contains the reciprocal pivot growth factor for the leading INFO columns of A.

  • INFO — INTEGER
    • = 0: successful exit, we ate 💅
    • < 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 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, but make it math. ✨

👑 Author 👑

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

🌈 subroutine cgbsvxx 🌈

subroutine cgbsvxx (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 RPVGRW, real,
    dimension( * ) BERR, integer N_ERR_BNDS, real, dimension( nrhs, * )
    ERR_BNDS_NORM, real, dimension( nrhs, * ) ERR_BNDS_COMP, integer
    NPARAMS, real, dimension( * ) PARAMS, complex, dimension( * ) WORK,
    real, dimension( * ) RWORK, integer INFO)

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

🎯 Purpose 🎯

CGBSVXX uses the LU factorization to compute the solution to a complex system of linear equations A * X = B, where A is an N-by-N matrix and X and B are N-by-NRHS matrices.

If you ask nicely, it returns both normwise and maximum componentwise error bounds. CGBSVXX will hand you 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 you get a warning instead. The relevant condition numbers are also calculated and returned. 🎀

CGBSVXX accepts user-provided factorizations and equilibration factors; peep the definitions of the FACT and EQUED options. Solving with refinement and using a factorization from a previous CGBSVXX 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 CGBSVXX would itself produce. No cap. 🚫🧢

📖 Description 📖

The following steps are performed:

  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'). It's giving transformation. 🦋

  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 honestly.

  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. She's persistent. 💪
  2. The system of equations is solved for X using the factored form of A.
  3. By default (so like, unless PARAMS(LA_LINRX_ITREF_I) is set to zero, bestie) the routine is gonna use iterative refinement to try and get a small error and small error bounds. Refinement calculates the residual to at least twice the working precision. It's giving thorough. 💅
  4. If equilibration was used, the matrix X gets premultiplied by diag(C) (when TRANS = 'N') or by diag(R) (when TRANS = 'T' or 'C') so that it solves the original system before equilibration. No cap. ✨

Some optional parameters are bundled up in the PARAMS array — very "everything in one bag" of them. These settings determine how refinement is performed, but honestly? The defaults are usually fine. If the defaults are acceptable to you, you can pass NPARAMS = 0, which stops the source code from even touching the PARAMS argument. Efficient queen behavior. 👑

🌸 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. Choose your own adventure:

  • = '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.
  • = 'N': The matrix A will be copied to AF and factored.
  • = 'E': The matrix A will be equilibrated if necessary, then copied to AF and factored.

TRANS

TRANS is CHARACTER*1

Specifies the form of the system of equations. Pick your fighter:

  • = 'N': A * X = B (No transpose)
  • = 'T': A**T * X = B (Transpose)
  • = 'C': A**H * X = B (Conjugate Transpose = 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 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. Leave it alone, it's fine.

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 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.

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).

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 = P*L*U as computed by CGETRF; 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 = P*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 = P*L*U of the equilibrated matrix A.

EQUED

EQUED is CHARACTER*1

Specifies the form of equilibration that was done. Vibe check on the scaling:

  • = '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. 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. 😭 Powers of the radix or it's not it, bestie.

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. 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 deal as R, literally the column twin.

B

B is 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 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.
Heads up: your B might come back looking different than you left it. That's not a bug, that's the scaling. 👀

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, 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

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. Tiny number, big drama, not always a crisis. 💀

RPVGRW

RPVGRW is REAL

Reciprocal pivot growth. On exit, this contains the reciprocal pivot growth factor norm(A)/norm(U). The 'max absolute element' 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 CGESVX, this quantity is returned in RWORK(1). 🫶

BERR

BERR is REAL 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. 🌈

ERR_BNDS_NORM

ERR_BNDS_NORM is REAL 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. 👑

Okay so the first index in ERR_BNDS_NORM(i,:)? That's literally the ith right-hand side, bestie. 💖

The second index in ERR_BNDS_NORM(:,err) contains these three fields, and they are giving:

  • err = 1 'Trust/don't trust' boolean. Trust the answer if the reciprocal condition number is less than the threshold sqrt(n) * slamch('Epsilon'). Vibe check ✨
  • err = 2 'Guaranteed' 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) * slamch('Epsilon'). This error bound should only be trusted if the previous boolean is true. No cap, only trust it if the boolean said so. 💅
  • err = 3 Reciprocal condition number: Estimated normwise reciprocal condition number. Compared with the threshold sqrt(n) * slamch('Epsilon') to determine if the error estimate is 'guaranteed'. 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. Math girlies, this is for you. 🧮

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

ERR_BNDS_COMP

ERR_BNDS_COMP is REAL 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))

It's giving componentwise realness. 💅

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 'Trust/don't trust' boolean. Trust the answer if the reciprocal condition number is less than the threshold sqrt(n) * slamch('Epsilon'). Same vibe check, different day. ✨
  • err = 2 'Guaranteed' 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) * slamch('Epsilon'). This error bound should only be trusted if the previous boolean is true. Don't skip the boolean, bestie. 💀
  • err = 3 Reciprocal condition number: Estimated componentwise reciprocal condition number. Compared with the threshold sqrt(n) * slamch('Epsilon') to determine if the error estimate is 'guaranteed'. 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. Look at the diag(x) glow-up. ✨

See Lapack Working Note 165 for further details and extra cautions. Read it, periodt. 📚

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. So if you're not about that life, it just uses defaults. 🍓

PARAMS

PARAMS is REAL array, dimension NPARAMS. Specifies algorithm parameters. If an entry is < 0.0, then that entry will be filled with default value used for that parameter. Only positions up to NPARAMS are accessed; defaults are used for higher-numbered parameters. Negative? That's just asking for the default, no cap. 💅

  • PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative refinement or not. Default: 1.0
    • = 0.0: No refinement is performed, and no error bounds are computed. No refinement, no bounds. Simple. 🙈
    • = 1.0: Use the double-precision refinement algorithm, possibly with doubled-single computations if the compilation environment does not support DOUBLE PRECISION. Double the precision, double the fun. ✨
    • (other values are reserved for future use) Future tea. 🫖
  • PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual computations allowed for refinement. Default: 10 Aggressive: Set to 100 to permit convergence using approximate factorizations or factorizations other than LU. 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. Warning: if you go off-script with factorization, the guarantees might ghost you. 💀
  • PARAMS(LA_LINRX_CWISE_I = 3) : 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) Default is on, so it's trying its best for componentwise accuracy. 💖

WORK

WORK is COMPLEX array, dimension (2*N). Just a workspace, doing its thing. 🛠️

RWORK

RWORK is REAL array, dimension (2*N). Real workspace, keeping it real. 💯

INFO

INFO is INTEGER. Here's the tea on what it means: 🫖

  • = 0: Successful exit. The solution to every right-hand side is guaranteed. Slay! All guaranteed. 👑
  • < 0: If INFO = -i, the i-th argument had an illegal value. Oops, bad argument. 😭
  • > 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. Singular matrix alert! No solution for you. 💀
  • = 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. Basically, J is the first problematic right-hand side. Check the arrays for the full story. 🔍

Author

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

Author Generated automatically by Doxygen for LAPACK from the source code. Shoutout to the LAPACK icons who wrote this! 💖

And that's a wrap! This page is from LAPACK version 3.11.0, dated Sun Nov 27 2022. Periodt. 💅

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