complex16PTsolve(3)

📦
version 3.11.0 · 2 versions
, in debian bookworm

🌸 NAME 🌸

complex16PTsolve — the complex16 girlies of the PT-solve world 💅

✨ SYNOPSIS ✨

🎀 Functions

  • subroutine zptsv (N, NRHS, D, E, B, LDB, INFO) — ZPTSV computes the solution to system of linear equations A * X = B for PT matrices
  • subroutine zptsvx (FACT, N, NRHS, D, E, DF, EF, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK, INFO) — ZPTSVX computes the solution to system of linear equations A * X = B for PT matrices

🦋 Detailed Description 🦋

This is the group of complex16 solve driver functions for PT matrices. yes bestie, a whole lil squad of them. periodt.

👑 Function Documentation 👑

🍓 subroutine zptsv

subroutine zptsv (integer N, integer NRHS, double precision, dimension( * )
    D, complex*16, dimension( * ) E, complex*16, dimension( ldb, * ) B,
    integer LDB, integer INFO)

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

💜 Purpose

ZPTSV computes the solution to a complex system of linear equations

A*X = B,

where A is an N-by-N Hermitian positive definite tridiagonal matrix, and X and B are N-by-NRHS matrices.

A is factored as A = L*D*L**H, and the factored form of A is then used to solve the system of equations.

🌈 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**H.
  • E
    • E is COMPLEX*16 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**H factorization of A. E can also be regarded as the superdiagonal of the unit bidiagonal factor U from the U**H*D*U factorization of A.
  • B
    • B is COMPLEX*16 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.

🫶 Author

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

🧚 subroutine zptsvx

subroutine zptsvx (character FACT, integer N, integer NRHS, double
    precision, dimension( * ) D, complex*16, dimension( * ) E, double
    precision, dimension( * ) DF, complex*16, dimension( * ) EF, 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)

ZPTSVX computes the solution to system of linear equations A * X = B for PT matrices. the overachiever of the family fr fr.

💜 Purpose

ZPTSVX uses the factorization A = L*D*L**H to compute the solution to a complex system of linear equations A*X = B, where A is an N-by-N Hermitian 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. it's giving thorough.

🌸 Description

The following steps are performed:

  1. If FACT = 'N', the matrix A is factored as A = L*D*L**H, where L is a unit lower bidiagonal matrix and D is diagonal. The factorization can also be regarded as having the form A = U**H*D*U.
  2. 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.
  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 the matrix A is 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 COMPLEX*16 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**H 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**H factorization of A.
  • EF
    • EF is COMPLEX*16 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**H 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**H factorization of A.
  • 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 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 COMPLEX*16 array, dimension (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: 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.

🫶 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.

no cap, a robot wrote this section and honestly? it ate. 🤖✨

okay bestie that's the whole page 🫶 this was LAPACK, version 3.11.0, dated Sun Nov 27 2022, and honestly the way this manpage showed up and understood the assignment?? iconic. go solve something Hermitian positive definite tridiagonal, you've got this 💅👑

📦 where this page comes from