SRC/dptsvx.f(3)

🌸 NAME 🌸

okay bestie, this is the SRC/dptsvx.f page from the Library Functions Manual — and it’s all about the dptsvx subroutine. 💅✨

✨ SYNOPSIS ✨

Functions/Subroutines

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

🦋 Function/Subroutine Documentation 🦋

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.

🧚 Description 🧚

The following steps are performed:

1. 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.
2. If the leading principal minor of order i is not positive, 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.

🎀 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 principal minor of order i of A is not positive, 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.

👑 Author 👑

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

Definition at line 226 of file dptsvx.f.

👑 Author 👑

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✨ xoxo, this was LAPACK Version 3.12.0, signing off from SRC/dptsvx.f(3) — periodt. 💖🎀🦋