ZGTSVX(3)

💅 NAME 💅

ok so this is the manual page for /home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/zgtsvx.f — and yes bestie, that's a whole file path as a NAME, we don't question it, we just vibe. It's part of the LAPACK Library Functions Manual, so buckle in 🎀

✨ SYNOPSIS ✨

🌸 Functions/Subroutines

subroutine ZGTSVX (fact, trans, n, nrhs, dl, d, du, dlf, df, duf, du2,
    ipiv, b, ldb, x, ldx, rcond, ferr, berr, work, rwork, info)

ZGTSVX computes the solution to system of linear equations A * X = B for GT matrices. that's it, that's the whole flex 💅

🦋 Function/Subroutine Documentation 🦋

👑 subroutine ZGTSVX

subroutine ZGTSVX (character fact, character trans, integer n, integer nrhs,
    complex*16, dimension( * ) dl, complex*16, dimension( * ) d, complex*16,
    dimension( * ) du, complex*16, dimension( * ) dlf, complex*16,
    dimension( * ) df, complex*16, dimension( * ) duf, complex*16,
    dimension( * ) du2, integer, dimension( * ) ipiv, 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)

ZGTSVX computes the solution to system of linear equations A * X = B for GT matrices. iconic behaviour, no notes 🔥

🌈 Purpose:

!>
!> ZGTSVX 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 tridiagonal matrix of order N and X and B are N-by-NRHS
!> matrices.
!>
!> Error bounds on the solution and a condition estimate are also
!> provided.
!>

so basically: tridiagonal matrix A of order N, right-hand sides X and B that are N-by-NRHS, and it handles A * X = B, A**T * X = B, or A**H * X = B. and it also serves you error bounds AND a condition estimate on the side. the full combo meal fr fr 🍓

🧚 Description:

!>
!> The following steps are performed:
!>
!> 1. If FACT = 'N', the LU decomposition is used to factor the matrix A
!>    as A = L * U, where L is a product of permutation and unit lower
!>    bidiagonal matrices and U is upper triangular with nonzeros in
!>    only the main diagonal and first two 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.
!>

ok so here's the step-by-step rundown, bestie 👇

if FACT = 'N', the LU decomposition factors A as A = L * U, where L is a product of permutation and unit lower bidiagonal matrices, and U is upper triangular with nonzeros in only the main diagonal and first two superdiagonals.
if some U(i,i)=0, so U is exactly singular, the routine bounces with INFO = i. otherwise the factored form of A gets used to estimate the condition number of A. and listen — if the reciprocal of the condition number is less than machine precision, INFO = N+1 comes back as a warning, but the routine still goes on to solve for X and compute error bounds like described below. it doesn't quit on you, that's loyalty 🫶
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. ✨ glow up ✨

💜 Parameters

FACT — FACT is CHARACTER*1. says whether or not the factored form of A was supplied on entry.
- = 'F': DLF, DF, DUF, DU2, and IPIV contain the factored form of A; DL, D, DU, DLF, DF, DUF, DU2 and IPIV will not be modified.
- = 'N': the matrix will be copied to DLF, DF, and DUF and factored.
TRANS — TRANS is CHARACTER*1. says which form of the system of equations we're on:
- = 'N': A * X = B (No transpose)
- = 'T': A**T * X = B (Transpose)
- = 'C': A**H * X = B (Conjugate transpose)
N — N is INTEGER. the order of the matrix A. N >= 0.
NRHS — NRHS is INTEGER. the number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.
DL — DL is COMPLEX*16 array, dimension (N-1). the (n-1) subdiagonal elements of A.
D — D is COMPLEX*16 array, dimension (N). the n diagonal elements of A.
DU — DU is COMPLEX*16 array, dimension (N-1). the (n-1) superdiagonal elements of A.
DLF — DLF is COMPLEX*16 array, dimension (N-1). if FACT = 'F', then DLF is an input argument and on entry contains the (n-1) multipliers that define the matrix L from the LU factorization of A as computed by ZGTTRF. if FACT = 'N', then DLF is an output argument and on exit contains the (n-1) multipliers that define the matrix L from the LU factorization of A.
DF — DF is COMPLEX*16 array, dimension (N). if FACT = 'F', then DF is an input argument and on entry contains the n diagonal elements of the upper triangular matrix U from the LU factorization of A. if FACT = 'N', then DF is an output argument and on exit contains the n diagonal elements of the upper triangular matrix U from the LU factorization of A.
DUF — DUF is COMPLEX*16 array, dimension (N-1). if FACT = 'F', then DUF is an input argument and on entry contains the (n-1) elements of the first superdiagonal of U. if FACT = 'N', then DUF is an output argument and on exit contains the (n-1) elements of the first superdiagonal of U.
DU2 — DU2 is COMPLEX*16 array, dimension (N-2). if FACT = 'F', then DU2 is an input argument and on entry contains the (n-2) elements of the second superdiagonal of U. if FACT = 'N', then DU2 is an output argument and on exit contains the (n-2) elements of the second superdiagonal of U.
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 LU factorization of A as computed by ZGTTRF. if FACT = 'N', then IPIV is an output argument and on exit contains the pivot indices from the LU factorization of A; row i of the matrix was interchanged with row IPIV(i). IPIV(i) will always be either i or i+1; IPIV(i) = i indicates a row interchange was not required.
B — B is COMPLEX*16 array, dimension (LDB,NRHS). the N-by-NRHS right hand side matrix B.
LDB — LDB is INTEGER. the leading dimension of the array B. LDB >= max(1,N).
X — X is COMPLEX*16 array, dimension (LDX,NRHS). if INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X. 👑
LDX — LDX is INTEGER. the leading dimension of the array X. LDX >= max(1,N).
RCOND — RCOND is DOUBLE PRECISION. the estimate of the reciprocal condition number of the matrix A. 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.
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).
WORK — WORK is COMPLEX*16 array, dimension (2*N).
RWORK — RWORK is DOUBLE PRECISION 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 not been completed unless i = N, 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 291 of file zgtsvx.f. 🔥

🌟 Author

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ok that's the whole page bestie, go forth and factor those tridiagonal matrices 💖✨

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