realGTsolve(3)

📦
version 3.11.0 · 2 versions
, in debian bookworm

✨ realGTsolve(3) — LAPACK ✨

okay bestie, welcome to the GT matrix solve squad 💖 this is the group of real solve driver functions for GT matrices. no cap, we're doing linear algebra and we're doing it cute. periodt.

🌸 NAME 🌸

realGTsolve - real

🎀 SYNOPSIS 🎀

🦋 Functions 🦋

  • subroutine sgtsv (N, NRHS, DL, D, DU, B, LDB, INFO) — SGTSV computes the solution to system of linear equations A * X = B for GT matrices.
  • subroutine sgtsvx (FACT, TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, IWORK, INFO) — SGTSVX computes the solution to system of linear equations A * X = B for GT matrices.

💖 Detailed Description 💖

This is the group of real solve driver functions for GT matrices. that's the whole vibe, bestie. it's giving "we solve tridiagonal systems and we look good doing it". 💅

🔥 Function Documentation 🔥

⭐ subroutine sgtsv ⭐

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

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

🎯 Purpose 🎯

SGTSV 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. iykyk, that's a little trick, no cap.

📋 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 REAL array, dimension (N-1). On entry, DL must contain the (n-1) sub-diagonal elements of A.

On exit, DL is overwritten by the (n-2) elements of the second super-diagonal of the upper triangular matrix U from the LU factorization of A, in DL(1), ..., DL(n-2).

  • D — D is REAL 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 REAL array, dimension (N-1). On entry, DU must contain the (n-1) super-diagonal elements of A.

On exit, DU is overwritten by the (n-1) elements of the first super-diagonal of U.

  • B — B is REAL array, dimension (LDB,NRHS). On entry, the N by NRHS matrix of 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 sgtsvx ⭐

subroutine sgtsvx (character FACT, character TRANS, integer N, integer NRHS, real, dimension( * ) DL, real, dimension( * ) D, real, dimension( * ) DU, real, dimension( * ) DLF, real, dimension( * ) DF, real, dimension( * ) DUF, real, dimension( * ) DU2, integer, dimension( * ) IPIV, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldx, * ) X, integer LDX, real RCOND, real, dimension( * ) FERR, real, dimension( * ) BERR, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)

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

🎯 Purpose 🎯

SGTSVX 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. obsessed, honestly. 💅

📝 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.

📋 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 = 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 REAL array, dimension (N-1). The (n-1) subdiagonal elements of A.
  • D — D is REAL array, dimension (N). The n diagonal elements of A.
  • DU — DU is REAL array, dimension (N-1). The (n-1) superdiagonal elements of A.
  • DLF — DLF is REAL 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 SGTTRF.
    • 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 REAL 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 REAL 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 REAL 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 SGTTRF.
    • 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 REAL 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 REAL 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 REAL array, dimension (3*N).
  • IWORK — IWORK is INTEGER array, dimension (N).
  • INFO —

Ok so back to INFO — this one is the receipts 🧾, bestie. It's an INTEGER, and what it tells you depends on its sign, so pay attention fr fr:

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.

Break it down with me 💅:

  • = 0 — successful exit ✨ she did it, no notes, we love to see it.
  • < 0 — if INFO = -i, then the i-th argument had an illegal value 🚨. You passed something cursed, bestie, that's on you.
  • > 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 "girl the math said no" 😭.
    • = 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… don't panic, but also don't @ me 🌸.
⚠️ bestie warning: RCOND = 0 means the error bounds straight up do not exist. Do not go using them. Periodt.

Author

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

Like, the whole academic Avengers assembled for this one 🫶👑

👑 Author

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And that's the whole page, bestie!! 🎀 This little icon was served to you fresh from the LAPACK source code via Doxygen, Version 3.11.0, dated Sun Nov 27 2022 — so if anything here is giving slightly vintage, that's why 💖. Now go forth and realGTsolve responsibly, mwah 💅✨🦋

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