doubleGTcomputational(3)

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version 3.11.0 · 2 versions
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✨ doubleGTcomputational(3) ✨ — LAPACK's tridiagonal double-precision bestie page 💅

okay so bestie, this is the doubleGTcomputational(3) manual, part 1. It's giving LAPACK, it's giving tridiagonal matrices, it's giving computational double precision. Let's get into it. 🦋

NAME 💖

doubleGTcomputational — double

SYNOPSIS ✨

Functions 🎀

       subroutine dgtcon (NORM, N, DL, D, DU, DU2, IPIV, ANORM, RCOND, WORK,
           IWORK, INFO)
           DGTCON
       subroutine dgtrfs (TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2, IPIV,
           B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO)
           DGTRFS
       subroutine dgttrf (N, DL, D, DU, DU2, IPIV, INFO)
           DGTTRF
       subroutine dgttrs (TRANS, N, NRHS, DL, D, DU, DU2, IPIV, B, LDB, INFO)
           DGTTRS
       subroutine dgtts2 (ITRANS, N, NRHS, DL, D, DU, DU2, IPIV, B, LDB)
           DGTTS2 solves a system of linear equations with a tridiagonal matrix
           using the LU factorization computed by sgttrf.
  • dgtcon — DGTCON
  • dgtrfs — DGTRFS
  • dgttrf — DGTTRF
  • dgttrs — DGTTRS
  • dgtts2 — DGTTS2 solves a system of linear equations with a tridiagonal matrix using the LU factorization computed by sgttrf.

Detailed Description 🌈

okay so this is literally the group of double computational functions for GT matrices. Tridiagonal besties, you know the vibe. 💅

Function Documentation 🎀

subroutine dgtcon — DGTCON ✨

   subroutine dgtcon (character NORM, integer N, double precision, dimension( *
       ) DL, double precision, dimension( * ) D, double precision, dimension( *
       ) DU, double precision, dimension( * ) DU2, integer, dimension( * )
       IPIV, double precision ANORM, double precision RCOND, double precision,
       dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)
       DGTCON

Purpose 💅

okay so DGTCON estimates the reciprocal of the condition number of a real tridiagonal matrix A using the LU factorization as computed by DGTTRF.

An estimate is obtained for norm(inv(A)), and the reciprocal of the condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))). Periodt.

Parameters 🫶

  • NORM — NORM is CHARACTER*1. Specifies whether the 1-norm condition number or the infinity-norm condition number is required:
    • = '1' or 'O': 1-norm;
    • = 'I': Infinity-norm.
  • N — N is INTEGER. The order of the matrix A. N >= 0.
  • DL — DL is DOUBLE PRECISION array, dimension (N-1). The (n-1) multipliers that define the matrix L from the LU factorization of A as computed by DGTTRF.
  • D — D is DOUBLE PRECISION array, dimension (N). The n diagonal elements of the upper triangular matrix U from the LU factorization of A.
  • DU — DU is DOUBLE PRECISION array, dimension (N-1). The (n-1) elements of the first superdiagonal of U.
  • DU2 — DU2 is DOUBLE PRECISION array, dimension (N-2). The (n-2) elements of the second superdiagonal of U.
  • IPIV — IPIV is INTEGER array, dimension (N). The pivot indices; for 1 <= i <= n, 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.
  • ANORM — ANORM is DOUBLE PRECISION. If NORM = '1' or 'O', the 1-norm of the original matrix A. If NORM = 'I', the infinity-norm of the original matrix A.
  • RCOND — RCOND is DOUBLE PRECISION. The reciprocal of the condition number of the matrix A, computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an estimate of the 1-norm of inv(A) computed in this routine.
  • WORK — WORK is DOUBLE PRECISION array, dimension (2*N).
  • IWORK — 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

Author ⭐

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

subroutine dgtrfs — DGTRFS ✨

   subroutine dgtrfs (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, dimension( * )
       FERR, double precision, dimension( * ) BERR, double precision,
       dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)
       DGTRFS

Purpose 💅

DGTRFS improves the computed solution to a system of linear equations when the coefficient matrix is tridiagonal, and provides error bounds and backward error estimates for the solution. It's the aftercare for your linear solve, bestie. 🫶

Parameters 🫶

  • 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 DOUBLE PRECISION array, dimension (N-1). The (n-1) subdiagonal elements of A.
  • D — D is DOUBLE PRECISION array, dimension (N). The diagonal elements of A.
  • DU — DU is DOUBLE PRECISION array, dimension (N-1). The (n-1) superdiagonal elements of A.
  • DLF — DLF is DOUBLE PRECISION array, dimension (N-1). The (n-1) multipliers that define the matrix L from the LU factorization of A as computed by DGTTRF.
  • DF — DF is DOUBLE PRECISION array, dimension (N). 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). The (n-1) elements of the first superdiagonal of U.
  • DU2 — DU2 is DOUBLE PRECISION array, dimension (N-2). The (n-2) elements of the second superdiagonal of U.
  • IPIV — IPIV is INTEGER array, dimension (N). The pivot indices; for 1 <= i <= n, 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 DOUBLE PRECISION array, dimension (LDB,NRHS). The 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). On entry, the solution matrix X, as computed by DGTTRS. On exit, the improved solution matrix X.
  • LDX — LDX is INTEGER. The leading dimension of the array X. LDX >= max(1,N).
  • 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.
  • 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 (3*N).
  • IWORK — 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

Internal Parameters 🍓

ITMAX is the maximum number of steps of iterative refinement.

Author ⭐

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

subroutine dgttrf — DGTTRF ✨

   subroutine dgttrf (integer N, double precision, dimension( * ) DL, double
       precision, dimension( * ) D, double precision, dimension( * ) DU, double
       precision, dimension( * ) DU2, integer, dimension( * ) IPIV, integer
       INFO)
       DGTTRF

Purpose 💅

DGTTRF computes an LU factorization of a real tridiagonal matrix A using elimination with partial pivoting and row interchanges.

The factorization has the form

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. It's giving factorization realness. 🔥

Parameters 🫶

  • N — N is INTEGER. The order of the matrix A.
  • DL — DL is DOUBLE PRECISION 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-1) multipliers that define the matrix L from the LU factorization of A.
  • D — D is DOUBLE PRECISION array, dimension (N). On entry, D must contain the diagonal elements of A. On exit, D is overwritten by the n diagonal elements of the upper triangular matrix U from the LU factorization of A.
  • DU — DU is DOUBLE PRECISION 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.
  • DU2 — DU2 is DOUBLE PRECISION array, dimension (N-2). On exit, DU2 is overwritten by the (n-2) elements of the second super-diagonal of U.
  • IPIV — IPIV is INTEGER array, dimension (N). The pivot indices; for 1 <= i <= n, 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.
  • INFO —
  • = 0: successful exit, we love that for you 💖
  • < 0: if INFO = -k, the k-th argument had an illegal value 🚨
  • > 0: if INFO = k, U(k,k) is exactly zero. The factorization has been completed, but the factor U is exactly singular, and division by zero will occur if it is used to solve a system of equations. 💀 bestie this one is a problem, not a vibe

🌸 Author 🌸

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

dgttrs 🦋

subroutine dgttrs (character TRANS, integer N, integer NRHS, double
    precision, dimension( * ) DL, double precision, dimension( * ) D, double
    precision, dimension( * ) DU, double precision, dimension( * ) DU2,
    integer, dimension( * ) IPIV, double precision, dimension( ldb, * ) B,
    integer LDB, integer INFO)

DGTTRS

✨ Purpose ✨

DGTTRS solves one of the systems of equations

   A*X = B  or  A**T*X = B,

with a tridiagonal matrix A using the LU factorization computed by DGTTRF. 🔥

💅 Parameters 💅

  • TRANS — TRANS is CHARACTER*1. Specifies the form of the system of equations. It's giving which math are we doing today:
    • = 'N': A * X = B (No transpose)
    • = 'T': A**T* X = B (Transpose)
    • = 'C': A**T* X = B (Conjugate transpose = Transpose)
  • N — N is INTEGER. The order of the matrix A. 👑
  • 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 DOUBLE PRECISION array, dimension (N-1). The (n-1) multipliers that define the matrix L from the LU factorization of A. 🍓
  • D — D is DOUBLE PRECISION array, dimension (N). The n diagonal elements of the upper triangular matrix U from the LU factorization of A.
  • DU — DU is DOUBLE PRECISION array, dimension (N-1). The (n-1) elements of the first super-diagonal of U.
  • DU2 — DU2 is DOUBLE PRECISION array, dimension (N-2). The (n-2) elements of the second super-diagonal of U. two whole super-diagonals, iconic 🎀
  • IPIV — IPIV is INTEGER array, dimension (N). The pivot indices; for 1 <= i <= n, 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 DOUBLE PRECISION array, dimension (LDB,NRHS). On entry, the matrix of right hand side vectors B. On exit, B is overwritten by the solution vectors X. ⚠️ not me leaving with the same array and different numbers in it, that's just how this function rolls
  • 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 🚩

🌸 Author 🌸

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

dgtts2 🧚

subroutine dgtts2 (integer ITRANS, integer N, integer NRHS, double
    precision, dimension( * ) DL, double precision, dimension( * ) D, double
    precision, dimension( * ) DU, double precision, dimension( * ) DU2,
    integer, dimension( * ) IPIV, double precision, dimension( ldb, * ) B,
    integer LDB)

DGTTS2 solves a system of linear equations with a tridiagonal matrix using the LU factorization computed by sgttrf.

✨ Purpose ✨

DGTTS2 solves one of the systems of equations

   A*X = B  or  A**T*X = B,

with a tridiagonal matrix A using the LU factorization computed by DGTTRF. 💜

💅 Parameters 💅

  • ITRANS — ITRANS is INTEGER. Specifies the form of the system of equations. yep, this one takes an integer instead of a letter, different function different personality:
    • = 0: A * X = B (No transpose)
    • = 1: A**T* X = B (Transpose)
    • = 2: A**T* X = B (Conjugate transpose = Transpose)
  • N — N is INTEGER. The order of the matrix A. 👑
  • 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 DOUBLE PRECISION array, dimension (N-1). The (n-1) multipliers that define the matrix L from the LU factorization of A. 🍓
  • D — D is DOUBLE PRECISION array, dimension (N). The n diagonal elements of the upper triangular matrix U from the LU factorization of A.
  • DU — DU is DOUBLE PRECISION array, dimension (N-1). The (n-1) elements of the first super-diagonal of U.
  • DU2 — DU2 is DOUBLE PRECISION array, dimension (N-2). The (n-2) elements of the second super-diagonal of U. 🎀
  • IPIV — IPIV is INTEGER array, dimension (N). The pivot indices; for 1 <= i <= n, 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 DOUBLE PRECISION array, dimension (LDB,NRHS). On entry, the matrix of right hand side vectors B. On exit, B is overwritten by the solution vectors X. ⚠️
  • LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N).
heads up bestie: dgtts2 has no INFO argument. That's not me forgetting it, that's literally the signature. no error reporting era 😭

🌸 Author 🌸

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

💖 Author 💖

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and that's the whole page, bestie, you made it to the end 🫶 this was Version 3.11.0 of the docs, dated Sun Nov 27 2022, so go off and solve your tridiagonal systems responsibly 💅👑

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