DGTSVX(3)

🌸 NAME 🌸

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/dgtsvx.f

okay so bestie, this is a page from the Library Functions Manual (yes, the whole file path is the "name", we don't make the rules 💅) and it's literally all about DGTSVX — the routine that solves A * X = B for GT matrices (tridiagonal girls, rise up 🦋).

✨ SYNOPSIS ✨

🌈 Functions/Subroutines

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

DGTSVX computes the solution to system of linear equations A * X = B for GT matrices. that's the whole vibe, no cap 🔥

💜 Function/Subroutine Documentation 💜

🎀 subroutine DGTSVX

subroutine DGTSVX (character fact, character trans, integer n, integer nrhs,
    double precision, dimension( * ) dl, double precision, dimension( * ) d,
    double precision, dimension( * ) du, double precision, dimension( * )
    dlf, double precision, dimension( * ) df, double precision, dimension( *
    ) duf, double precision, dimension( * ) du2, integer, dimension( * )
    ipiv, double precision, dimension( ldb, * ) b, integer ldb, double
    precision, dimension( ldx, * ) x, integer ldx, double precision rcond,
    double precision, dimension( * ) ferr, double precision, dimension( * )
    berr, double precision, dimension( * ) work, integer, dimension( * )
    iwork, integer info)

DGTSVX computes the solution to system of linear equations A * X = B for GT matrices. she's the whole package fr fr 😭

🫶 Purpose

!> !> DGTSVX uses the LU factorization to compute the solution to a real !> system of linear equations A * X = B or A**T * 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 she takes your real tridiagonal matrix A (order N), and solves A * X = B or A**T * X = B, where X and B are N-by-NRHS matrices. and she doesn't just solve it — she also hands you error bounds and a condition estimate. overachiever behavior 👑

💖 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. !>

here's the step-by-step, bestie, in order, no skips:

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. she's compact like that 🧚
if some U(i,i)=0, so U is exactly singular, the routine returns with INFO = i. otherwise the factored form of A gets used to estimate the condition number of A. and 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. she does not quit on you 💀
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 phase ✨

🌟 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. !>

FACT is CHARACTER*1 and it specifies whether 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. (nothing gets touched, she's respectful like that 🫶)
= '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 = Transpose) !>

TRANS is CHARACTER*1 and 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 = Transpose)

🎀 N

!> N is INTEGER !> The order of the matrix A. N >= 0. !>

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. !>

NRHS is INTEGER, the number of right hand sides — i.e. the number of columns of the matrix B. NRHS >= 0.

🌈 DL

!> DL is DOUBLE PRECISION array, dimension (N-1) !> The (n-1) subdiagonal elements of A. !>

DL is DOUBLE PRECISION array, dimension (N-1) — the (n-1) subdiagonal elements of A.

⭐ D

!> D is DOUBLE PRECISION array, dimension (N) !> The n diagonal elements of A. !>

D is DOUBLE PRECISION array, dimension (N) — the n diagonal elements of A. main character energy, literally the main diagonal 👑

🍓 DU

!> DU is DOUBLE PRECISION array, dimension (N-1) !> The (n-1) superdiagonal elements of A. !>

DU is DOUBLE PRECISION array, dimension (N-1) — the (n-1) superdiagonal elements of A.

💜 DLF

!> DLF is DOUBLE PRECISION 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 DGTTRF. !> !> 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. !>

DLF is DOUBLE PRECISION array, dimension (N-1):

if FACT = 'F', DLF is an input argument, and on entry it contains the (n-1) multipliers that define the matrix L from the LU factorization of A as computed by DGTTRF.
if FACT = 'N', DLF is an output argument, and on exit it contains the (n-1) multipliers that define the matrix L from the LU factorization of A.

🧚 DF

!> DF is DOUBLE PRECISION 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. !>

DF is DOUBLE PRECISION array, dimension (N):

if FACT = 'F', DF is an input argument, and on entry it contains the n diagonal elements of the upper triangular matrix U from the LU factorization of A.
if FACT = 'N', DF is an output argument, and on exit it contains the n diagonal elements of the upper triangular matrix U from the LU factorization of A.

🫶 DUF

!> DUF is DOUBLE PRECISION 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. !>

DUF is DOUBLE PRECISION array, dimension (N-1):

if FACT = 'F', DUF is an input argument, and on entry it contains the (n-1) elements of the first superdiagonal of U.
if FACT = 'N', DUF is an output argument, and on exit it contains the (n-1) elements of the first superdiagonal of U.

🦋 DU2

!> DU2 is DOUBLE PRECISION 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. !>

DU2 is DOUBLE PRECISION array, dimension (N-2):

if FACT = 'F', DU2 is an input argument, and on entry it contains the (n-2) elements of the second superdiagonal of U.
if FACT = 'N', DU2 is an output argument, and on exit it 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 DGTTRF. !> !> 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. !>

IPIV is INTEGER array, dimension (N):

if FACT = 'F', IPIV is an input argument, and on entry it contains the pivot indices from the LU factorization of A as computed by DGTTRF.
if FACT = 'N', IPIV is an output argument, and on exit it 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. (no swap needed, she stayed put 💅)

💖 B

!> B is DOUBLE PRECISION array, dimension (LDB,NRHS) !> The N-by-NRHS right hand side matrix B. !>

B is DOUBLE PRECISION 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). !>

LDB is INTEGER, the leading dimension of the array B. LDB >= max(1,N).

⭐ X

!> X is DOUBLE PRECISION array, dimension (LDX,NRHS) !> If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X. !>

X is DOUBLE PRECISION array, dimension (LDX,NRHS) — if INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X. the moment we've all been waiting for 😭

🌈 LDX

!> LDX is INTEGER !> The leading dimension of the array X. LDX >= max(1,N). !>

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. !>

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. !>

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. she's humble about it 🥺

🧚 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). !>`

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 DOUBLE PRECISION array, dimension (3*N) !>

WORK is DOUBLE PRECISION array, dimension (3*N).

🦋 IWORK

!> IWORK is INTEGER array, dimension (N) !>

IWORK is INTEGER 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. !>

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. plot twist, she still delivers 🔥

👑 Author

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

Definition at line 290 of file dgtsvx.f.

🌸 Author (again, yes)

Generated automatically by Doxygen for LAPACK from the source code. a bot wrote this part, not me 🙈

and that's the tea, bestie 💅 — this page came straight out of LAPACK, Version 3.12.0, dated with the source tag /home/abui...RC/dgtsvx.f(3). go forth and solve your tridiagonal systems, you absolute icon 🎀✨🦋

📦 where this page comes from

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opensuse-leap 16.0: lapack-man 3.12.0-160000.2.2 · BSD-3-Clause
opensuse-leap 16.0: lapack-man 3.12.0-160000.2.2 · BSD-3-Clause
💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash-1017 on 2026-11-04: a language model's rewrite of the original, under its terms