realPTcomputational(3)

📦

version 3.11.0 · 2 versions

3.11.0 · debian bookworm (latest)
3.10.0 · ubuntu jammy

realPTcomputational(3) — the LAPACK PT girlies 💅

okay so bestie, welcome to realPTcomputational — this is a LAPACK manual page, and it's literally the whole group chat of real computational functions for PT matrices. yes, the name is a mouthful. no, we are not shortening it. it's iconic like that. 🎀

✨ NAME ✨

realPTcomputational — real

(yes bestie that's the whole name section. the man page said "real" and we're respecting it. short king behavior. 👑)

🌸 SYNOPSIS 🌸

Functions

subroutine sptcon (N, D, E, ANORM, RCOND, WORK, INFO)
    SPTCON
subroutine spteqr (COMPZ, N, D, E, Z, LDZ, WORK, INFO)
    SPTEQR
subroutine sptrfs (N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR, BERR, WORK, INFO)
    SPTRFS
subroutine spttrs (N, NRHS, D, E, B, LDB, INFO)
    SPTTRS
subroutine sptts2 (N, NRHS, D, E, B, LDB)
    SPTTS2 solves a tridiagonal system of the form AX=B using the L D LH
    factorization computed by spttrf.

five little subroutines. a whole friend group. let's meet them 🫶

🦋 Detailed Description 🦋

This is the group of real computational functions for PT matrices. that's it, that's the vibe. they all hang out together. periodt.

💖 Function Documentation 💖

`subroutine sptcon`

subroutine sptcon (integer N, real, dimension( * ) D, real, dimension( * ) E,
                   real ANORM, real RCOND, real, dimension( * ) WORK, integer INFO)

SPTCON

Purpose

SPTCON computes the reciprocal of the condition number (in the 1-norm) of a real symmetric positive definite tridiagonal matrix, using the factorization A = L*D*L**T or A = U**T*D*U computed by SPTTRF.

Norm(inv(A)) is computed by a direct method, and the reciprocal of the condition number is computed as

RCOND = 1 / (ANORM * norm(inv(A))).

so like… it's checking how ~shaky~ your matrix is. lowkey a vibe check for linear algebra. 😭

Parameters

N — N is INTEGER. The order of the matrix A. N >= 0.
D — D is REAL array, dimension (N). The n diagonal elements of the diagonal matrix D from the factorization of A, as computed by SPTTRF.
E — E is REAL array, dimension (N-1). The (n-1) off-diagonal elements of the unit bidiagonal factor U or L from the factorization of A, as computed by SPTTRF.
ANORM — ANORM is REAL. The 1-norm of the original matrix A.
RCOND — RCOND is REAL. The reciprocal of the condition number of the matrix A, computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is the 1-norm of inv(A) computed in this routine.
WORK — WORK is REAL 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.

Further Details

The method used is described in Nicholas J. Higham, 'Efficient Algorithms for Computing the Condition Number of a Tridiagonal Matrix', SIAM J. Sci. Stat. Comput., Vol. 7, No. 1, January 1986.

sir Higham really said "let me cook" and cooked. 🔥

`subroutine spteqr`

subroutine spteqr (character COMPZ, integer N, real, dimension( * ) D, real,
                   dimension( * ) E, real, dimension( ldz, * ) Z, integer LDZ,
                   real, dimension( * ) WORK, integer INFO)

SPTEQR

Purpose

SPTEQR computes all eigenvalues and, optionally, eigenvectors of a symmetric positive definite tridiagonal matrix by first factoring the matrix using SPTTRF, and then calling SBDSQR to compute the singular values of the bidiagonal factor.

This routine computes the eigenvalues of the positive definite tridiagonal matrix to high relative accuracy. What that means: if the eigenvalues range over many orders of magnitude in size, then the small eigenvalues and corresponding eigenvectors will be computed more accurately than, for example, with the standard QR method. tiny eigenvalues getting their moment. we love to see it. 🧚

The eigenvectors of a full or band symmetric positive definite matrix can also be found if SSYTRD, SSPTRD, or SSBTRD has been used to reduce this matrix to tridiagonal form. (The reduction to tridiagonal form, however, may preclude the possibility of obtaining high relative accuracy in the small eigenvalues of the original matrix, if these eigenvalues range over many orders of magnitude.) so like, the glow-up comes with a tiny asterisk bestie. read the fine print. 🙈

Parameters

COMPZ — COMPZ is CHARACTER*1.
- = 'N': Compute eigenvalues only.
- = 'V': Compute eigenvectors of original symmetric matrix also. Array Z contains the orthogonal matrix used to reduce the original matrix to tridiagonal form.
- = 'I': Compute eigenvectors of tridiagonal matrix also.
N — N is INTEGER. The order of the matrix. N >= 0.
D — D is REAL array, dimension (N). On entry, the n diagonal elements of the tridiagonal matrix. On normal exit, D contains the eigenvalues, in descending order.
E — E is REAL array, dimension (N-1). On entry, the (n-1) subdiagonal elements of the tridiagonal matrix. On exit, E has been destroyed. rip E, you served well 🫡
Z — Z is REAL array, dimension (LDZ, N). On entry, if COMPZ = 'V', the orthogonal matrix used in the reduction to tridiagonal form. On exit, if COMPZ = 'V', the orthonormal eigenvectors of the original symmetric matrix; if COMPZ = 'I', the orthonormal eigenvectors of the tridiagonal matrix. If INFO > 0 on exit, Z contains the eigenvectors associated with only the stored eigenvalues. If COMPZ = 'N', then Z is not referenced. (she's off the clock. not invited. 🚫)
LDZ — LDZ is INTEGER. The leading dimension of the array Z. LDZ >= 1, and if COMPZ = 'V' or 'I', LDZ >= max(1,N).
WORK — WORK is REAL array, dimension (4*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 Cholesky factorization of the matrix could not be performed because the i-th principal minor was not positive definite.
  - > N the SVD algorithm failed to converge; if INFO = N+i, i off-diagonal elements of the bidiagonal factor did not converge to zero.

Author

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

`subroutine sptrfs`

subroutine sptrfs (integer N, integer NRHS, real, dimension( * ) D, real,
                   dimension( * ) E, real, dimension( * ) DF, real,
                   dimension( * ) EF, real, dimension( ldb, * ) B, integer LDB,
                   real, dimension( ldx, * ) X, integer LDX, real,
                   dimension( * ) FERR, real, dimension( * ) BERR, real,
                   dimension( * ) WORK, integer INFO)

SPTRFS

Purpose

SPTRFS improves the computed solution to a system of linear equations when the coefficient matrix is symmetric positive definite and tridiagonal, and provides error bounds and backward error estimates for the solution.

basically it's the friend who looks at your answer and goes "okay but we can make it better" and then actually does. 💅

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 REAL array, dimension (N). The n diagonal elements of the tridiagonal matrix A.
E — E is REAL array, dimension (N-1). The (n-1) subdiagonal elements of the tridiagonal matrix A.
DF — DF is REAL array, dimension (N). The n diagonal elements of the diagonal matrix D from the factorization computed by SPTTRF.
EF — EF is REAL array, dimension (N-1). The (n-1) subdiagonal elements of the unit bidiagonal factor L from the factorization computed by SPTTRF.
B — B is REAL 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 REAL array, dimension (LDX,NRHS). On entry, the solution matrix X, as computed by SPTTRS. On exit, the improved solution matrix X. glow-up complete ✨
LDX — LDX is INTEGER. The leading dimension of the array X. LDX >= max(1,N).
FERR — FERR is REAL 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 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 (2*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. (yes it's giving "she keeps going until she's satisfied". relatable. 💀)

Author

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

`subroutine spttrs`

subroutine spttrs (integer N, integer NRHS, real, dimension( * ) D, real,
                   dimension( * ) E, real, dimension( ldb, * ) B, integer LDB,
                   integer INFO)

SPTTRS

Purpose

SPTTRS solves a tridiagonal system of the form

   A * X = B

using the L*D*L**T factorization of A computed by SPTTRF. D is a diagonal matrix specified in the vector D, L is a unit bidiagonal matrix whose subdiagonal is specified in the vector E, and X and B are N by NRHS matrices.

she solves. she factors. she serves. periodt. 👑

Parameters

N — N is INTEGER. The order of the tridiagonal 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 REAL array, dimension (N). The n diagonal elements of the diagonal matrix D from the L*D*L**T factorization of A.
E —

okay bestie we are literally mid-sentence on E's description for spttrs, which means part 2 is coming with the rest of this parameter, the rest of spttrs, and the whole sptts2 situation. don't go anywhere, we're not done, iykyk 💖🦋✨

E — okay bestie, E is a REAL array with dimension (N-1). It holds the (n-1) subdiagonal elements of the unit bidiagonal factor L from the L*D*L**T factorization of A. And plot twist: E can also be regarded as the superdiagonal of the unit bidiagonal factor U from the factorization A = U**T*D*U. ✨
B — B is a REAL array, dimension (LDB,NRHS). On entry, it's the right hand side vectors B for the system of linear equations. On exit, it's the solution vectors, X. It's giving main character energy 💅.
LDB — LDB is INTEGER. It's the leading dimension of the array B. Constraint: LDB >= max(1,N). No cap, fr fr.
INFO — INFO is INTEGER. = 0: successful exit, we love to see it. < 0: if INFO = -k, the k-th argument had an illegal value. Not the vibe, but okay.

Author 🫶

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

subroutine sptts2 ✨

subroutine sptts2 (integer N, integer NRHS, real, dimension( * ) D, real,
       dimension( * ) E, real, dimension( ldb, * ) B, integer LDB)

SPTTS2 solves a tridiagonal system of the form AX=B using the L D LH factorization computed by spttrf.

Purpose 🎯

So SPTTS2 solves a tridiagonal system that looks like A * X = B, using the L*D*L**T factorization of A that SPTTRF computed. D is a diagonal matrix specified in the vector D, L is a unit bidiagonal matrix whose subdiagonal is specified in the vector E, and X and B are N by NRHS matrices. Periodt.

Parameters 📝

N — N is INTEGER. It's the order of the tridiagonal matrix A. Constraint: N >= 0. Slay.
NRHS — NRHS is INTEGER. The number of right hand sides, i.e., the number of columns of the matrix B. Constraint: NRHS >= 0. Iconic.
D — D is a REAL array, dimension (N). It holds the n diagonal elements of the diagonal matrix D from the L*D*L**T factorization of A. 💅
E — E is a REAL array, dimension (N-1). It holds the (n-1) subdiagonal elements of the unit bidiagonal factor L from the L*D*L**T factorization of A. And yes, E can also be regarded as the superdiagonal of the unit bidiagonal factor U from the factorization A = U**T*D*U. ✨
B — B is a REAL array, dimension (LDB,NRHS). On entry, it's the right hand side vectors B for the system of linear equations. On exit, it's the solution vectors, X. Glow-up moment 💅.
LDB — LDB is INTEGER. The leading dimension of the array B. Constraint: LDB >= max(1,N). No cap.

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.

And that's the tea, bestie. This was LAPACK version 3.11.0, dated Sun Nov 27 2022, and the page is realPTcomputational(3). Stay slay. 💅✨

📦 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