doublePTsolve(3)

📦

version 3.11.0 · 2 versions

3.11.0 · debian bookworm (latest)
3.10.0 · ubuntu jammy

💖 doublePTsolve 💖 — bestie's guide to the LAPACK PT solve squad

okay so. you're in section 3 of the LAPACK manual, and this is doublePTsolve — the whole friend group of double precision solve driver routines for PT matrices. two functions, one vibe, infinite slay. let's get into it 🫶

✨ SYNOPSIS ✨

🌸 Functions 🌸

subroutine dptsv (N, NRHS, D, E, B, LDB, INFO) — DPTSV computes the solution to system of linear equations A * X = B for PT matrices
subroutine dptsvx (FACT, N, NRHS, D, E, DF, EF, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, INFO) — DPTSVX computes the solution to system of linear equations A * X = B for PT matrices

🦋 Detailed Description 🦋

This is the group of double solve driver functions for PT matrices. that's it, that's the whole vibe. two icons, one mission: get you an X. 💅

👑 Function Documentation 👑

⭐ subroutine dptsv ⭐

subroutine dptsv (integer N, integer NRHS, double precision, dimension( * )
    D, double precision, dimension( * ) E, double precision, dimension( ldb,
    * ) B, integer LDB, integer INFO)

DPTSV computes the solution to system of linear equations A * X = B for PT matrices.

💜 Purpose:

DPTSV computes the solution to a real system of linear equations A*X = B, where A is an N-by-N symmetric positive definite tridiagonal matrix, and X and B are N-by-NRHS matrices.

A is factored as A = L*D*L**T, and the factored form of A is then used to solve the system of equations. no cap, that's the whole trick. ✨

🎀 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.
D — D is DOUBLE PRECISION array, dimension (N). On entry, the n diagonal elements of the tridiagonal matrix A. On exit, the n diagonal elements of the diagonal matrix D from the factorization A = L*D*L**T.
E — E is DOUBLE PRECISION array, dimension (N-1). On entry, the (n-1) subdiagonal elements of the tridiagonal matrix A. On exit, the (n-1) subdiagonal elements of the unit bidiagonal factor L from the L*D*L**T factorization of A. (E can also be regarded as the superdiagonal of the unit bidiagonal factor U from the U**T*D*U factorization of A.) okay she's versatile 💅
B — B is DOUBLE PRECISION 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, the leading minor of order i is not positive definite, and the solution has not been computed. The factorization has not been completed unless i = N.

ngl if you get a positive INFO, the matrix said "not today bestie" — the leading minor of order i is not positive definite. no solution for you. 💀

🌈 Author 🌈

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

💫 subroutine dptsvx 💫

subroutine dptsvx (character FACT, integer N, integer NRHS, double
    precision, dimension( * ) D, double precision, dimension( * ) E, double
    precision, dimension( * ) DF, double precision, dimension( * ) EF,
    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 INFO)

DPTSVX computes the solution to system of linear equations A * X = B for PT matrices.

💜 Purpose:

DPTSVX uses the factorization A = L*D*L**T to compute the solution to a real system of linear equations A*X = B, where A is an N-by-N symmetric positive definite tridiagonal matrix and X and B are N-by-NRHS matrices.

Error bounds on the solution and a condition estimate are also provided. she does the extra credit too, main character energy fr fr 👑

🧚 Description:

The following steps are performed:

If FACT = 'N', the matrix A is factored as A = L*D*L**T, where L is a unit lower bidiagonal matrix and D is diagonal. The factorization can also be regarded as having the form A = U**T*D*U.
If the leading i-by-i principal minor is not positive definite, 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. (she warns you but she keeps going, that's growth 🌱)
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': On entry, DF and EF contain the factored form of A. D, E, DF, and EF will not be modified.
- = 'N': The matrix A will be copied to DF and EF and factored.
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 matrices B and X. NRHS >= 0.
D — D is DOUBLE PRECISION array, dimension (N). The n diagonal elements of the tridiagonal matrix A.
E — E is DOUBLE PRECISION array, dimension (N-1). The (n-1) subdiagonal elements of the tridiagonal matrix 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 diagonal matrix D from the L*D*L**T factorization of A. If FACT = 'N', then DF is an output argument and on exit contains the n diagonal elements of the diagonal matrix D from the L*D*L**T factorization of A.
EF — EF is DOUBLE PRECISION array, dimension (N-1). If FACT = 'F', then EF is an input argument and on entry contains the (n-1) subdiagonal elements of the unit bidiagonal factor L from the L*D*L**T factorization of A. If FACT = 'N', then EF is an output argument and on exit contains the (n-1) subdiagonal elements of the unit bidiagonal factor L from the L*D*L**T factorization of A.
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).
X — X is DOUBLE PRECISION array, dimension (LDX,NRHS). If INFO = 0 of 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 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 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).
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 DOUBLE PRECISION array, dimension (2*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: the leading minor of order i of A is not positive definite, so the factorization could not be completed, and the solution has not been 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.

lowkey the INFO = N+1 case is the plot twist: the matrix is singular to working precision but she still hands you the solution AND the error bounds, because sometimes the computed solution eats harder than RCOND would have you believe. iconic behavior honestly 🔥

🌈 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. (yes bestie, the docs wrote themselves, we just added the sparkle ✨)

and that's the whole page, periodt. this is LAPACK Version 3.11.0, dated Sun Nov 27 2022, and honestly? she ate. go solve your tridiagonal system, you've got this 💖🦋💅

📦 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