complexGTsolve(3)

📦

version 3.11.0 · 2 versions

3.11.0 · debian bookworm (latest)
3.10.0 · ubuntu jammy

💖 NAME 💖

complexGTsolve(3) — LAPACK, bestie. The NAME line says complexGTsolve - complex, and we're keeping it exactly like that. It's giving complex GT solve driver energy, no cap. 💅

✨ SYNOPSIS ✨

🌸 Functions 🌸

subroutine cgtsv (N, NRHS, DL, D, DU, B, LDB, INFO)

CGTSV computes the solution to system of linear equations A * X = B for GT matrices

subroutine cgtsvx (FACT, TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2,
    IPIV, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK, INFO)

CGTSVX computes the solution to system of linear equations A * X = B for GT matrices

🌈 Detailed Description 🌈

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

👑 Function Documentation 👑

🦋 subroutine cgtsv

subroutine cgtsv (integer N, integer NRHS, complex, dimension( * ) DL,
    complex, dimension( * ) D, complex, dimension( * ) DU, complex,
    dimension( ldb, * ) B, integer LDB, integer INFO)

CGTSV computes the solution to system of linear equations A * X = B for GT matrices

✨ Purpose ✨

CGTSV solves the equation

A*X = B,

where A is an N-by-N tridiagonal matrix, by Gaussian elimination with partial pivoting.

Note that the equation A**T *X = B may be solved by interchanging the order of the arguments DU and DL.

🎀 Parameters 🎀

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 array, dimension (N-1). On entry, DL must contain the (n-1) subdiagonal elements of A. On exit, DL is overwritten by the (n-2) elements of the second superdiagonal of the upper triangular matrix U from the LU factorization of A, in DL(1), ..., DL(n-2).
D — D is COMPLEX array, dimension (N). On entry, D must contain the diagonal elements of A. On exit, D is overwritten by the n diagonal elements of U.
DU — DU is COMPLEX array, dimension (N-1). On entry, DU must contain the (n-1) superdiagonal elements of A. On exit, DU is overwritten by the (n-1) elements of the first superdiagonal of U.
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, and the solution has not been computed. The factorization has not been completed unless i = N.

🌟 Author 🌟

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

🦋 subroutine cgtsvx

subroutine cgtsvx (character FACT, character TRANS, integer N, integer NRHS,
    complex, dimension( * ) DL, complex, dimension( * ) D, complex,
    dimension( * ) DU, complex, dimension( * ) DLF, complex, dimension( * )
    DF, complex, dimension( * ) DUF, complex, dimension( * ) DU2, integer,
    dimension( * ) IPIV, 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)

CGTSVX computes the solution to system of linear equations A * X = B for GT matrices

✨ Purpose ✨

CGTSVX 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.

💫 Description 💫

The following steps are performed:

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.
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.
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.

🎀 Parameters 🎀

FACT — FACT is CHARACTER*1. Specifies whether or not the factored form of A has been 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. 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 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 array, dimension (N-1). The (n-1) subdiagonal elements of A.
D — D is COMPLEX array, dimension (N). The n diagonal elements of A.
DU — DU is COMPLEX array, dimension (N-1). The (n-1) superdiagonal elements of A.
DLF — DLF is COMPLEX 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 CGTTRF. 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 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 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 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 CGTTRF. 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 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 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 REAL. 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 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), 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 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).
WORK — WORK is COMPLEX array, dimension (2*N)
RWORK — RWORK is REAL array, dimension (N)
INFO
INFO is an INTEGER — she's the receipts girl, the one who tells you what actually went down 👑
- = 0 — successful exit. We ate. 💅✨
- < 0 — if INFO = -i, then the i-th argument had an illegal value. bestie you passed her something cursed 😭
- > 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. it's giving "we tried, she's singular, we're going home" 💀
  - = 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. so like… she's technically singular but the math said "let her cook" 🍓

👑 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.

and that's the whole page bestie, no cap 🫶 this was Version 3.11.0, dated Sun Nov 27 2022, and it came to you from complexGTsolve(3) — go be the main character of your own linear algebra era, ily, bye 💜🦋

📦 where this page comes from

debian bookworm: liblapack-doc 3.11.0-2 · BSD-3-Clause
💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash on 2026-10-14: a language model's rewrite of the original, under its terms