complexPTcomputational(3)

📦
version 3.11.0 · 2 versions
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complexPTcomputational(3) — yes bestie, that's the page name, and it's a LAPACK one 💖 this is the whole glow-up crew of complex computational functions for PT matrices, and we're doing it in two parts because it's long long. Part 1, let's go ✨

✨ NAME ✨

complexPTcomputational — complex

🦋 SYNOPSIS 🦋

Functions

  • subroutine cptcon (N, D, E, ANORM, RCOND, RWORK, INFO) — CPTCON
  • subroutine cpteqr (COMPZ, N, D, E, Z, LDZ, WORK, INFO) — CPTEQR
  • subroutine cptrfs (UPLO, N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO) — CPTRFS
  • subroutine cpttrf (N, D, E, INFO) — CPTTRF
  • subroutine cpttrs (UPLO, N, NRHS, D, E, B, LDB, INFO) — CPTTRS
  • subroutine cptts2 (IUPLO, N, NRHS, D, E, B, LDB) — CPTTS2 solves a tridiagonal system of the form AX=B using the L D LH factorization computed by spttrf.

💖 Detailed Description 💖

Okay so, this is the group of complex computational functions for PT matrices. That's the whole vibe. Group chat of routines, one purpose: doing the actual work on PT matrices 🫶

👑 Function Documentation 👑

🌸 cptcon 🌸

subroutine cptcon (integer N, real, dimension( * ) D, complex, dimension( * ) E, real ANORM, real RCOND, real, dimension( * ) RWORK, integer INFO)

CPTCON

💅 Purpose 💅

CPTCON computes the reciprocal of the condition number (in the 1-norm) of a complex Hermitian positive definite tridiagonal matrix, using the factorization A = L*D*L**H or A = U**H*D*U computed by CPTTRF.

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

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

🎀 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 CPTTRF.

  • E

E is COMPLEX 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 CPTTRF.

  • 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.

  • RWORK

RWORK 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.

🦋 cpteqr 🦋

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

CPTEQR

💅 Purpose 💅

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

This routine computes the eigenvalues of the positive definite tridiagonal matrix to high relative accuracy. Which 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. Iconic behaviour, honestly 👑

The eigenvectors of a full or band positive definite Hermitian matrix can also be found if CHETRD, CHPTRD, or CHBTRD has been used to reduce this matrix to tridiagonal form. (But heads up bestie: the reduction to tridiagonal form 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.)

🎀 Parameters 🎀

  • COMPZ

COMPZ is CHARACTER*1

  • = 'N': Compute eigenvalues only.
  • = 'V': Compute eigenvectors of original Hermitian matrix also. Array Z contains the unitary 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. 💀 gone. reduced to atoms.

  • Z

Z is COMPLEX array, dimension (LDZ, N) On entry, if COMPZ = 'V', the unitary matrix used in the reduction to tridiagonal form. On exit, if COMPZ = 'V', the orthonormal eigenvectors of the original Hermitian 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.

  • 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.

🌈 cptrfs 🌈

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

CPTRFS

💅 Purpose 💅

CPTRFS improves the computed solution to a system of linear equations when the coefficient matrix is Hermitian positive definite and tridiagonal, and provides error bounds and backward error estimates for the solution. It's the glow-up pass on your answer ✨

🎀 Parameters 🎀

  • UPLO

UPLO is CHARACTER*1 Specifies whether the superdiagonal or the subdiagonal of the tridiagonal matrix A is stored and the form of the factorization:

  • = 'U': E is the superdiagonal of A, and A = U**H*D*U;
  • = 'L': E is the subdiagonal of A, and A = L*D*L**H. (The two forms are equivalent if A is real.)
  • 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 real diagonal elements of the tridiagonal matrix A.

  • E

E is COMPLEX array, dimension (N-1) — the (n-1) off-diagonal elements of the tridiagonal matrix A (see UPLO).

  • DF

DF is REAL array, dimension (N) — the n diagonal elements of the diagonal matrix D from the factorization computed by CPTTRF.

  • EF

EF is COMPLEX array, dimension (N-1) — the (n-1) off-diagonal elements of the unit bidiagonal factor U or L from the factorization computed by CPTTRF (see UPLO).

  • B

B is COMPLEX 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 COMPLEX array, dimension (LDX,NRHS) On entry, the solution matrix X, as computed by CPTTRS. 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 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 COMPLEX array, dimension (N)

  • RWORK

RWORK is REAL 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.

🧚 cpttrf 🧚

subroutine cpttrf (integer N, real, dimension( * ) D, complex, dimension( * ) E, integer INFO)

CPTTRF

💅 Purpose 💅

CPTTRF computes the L*D*L**H factorization of a complex Hermitian positive definite tridiagonal matrix A. The factorization may also be regarded as having the form A = U**H *D*U.

🎀 Parameters 🎀

  • N
…aaand that's the part 1 cutoff, bestie — the rest of cpttrf and the whole rest of the crew is waiting in part 2 🫶

N

N is INTEGER. It's the order of the matrix A. And yes, N >= 0. Periodt. ✨

D

D is a REAL array, dimension (N). On entry, it's holding the n diagonal elements of the tridiagonal matrix A. On exit, it's serving the n diagonal elements of the diagonal matrix D from the L*D*L**H factorization of A. The glow-up is real. 💅

E

E is a COMPLEX array, dimension (N-1). On entry, it's the (n-1) subdiagonal elements of the tridiagonal matrix A. On exit, it's the (n-1) subdiagonal elements of the unit bidiagonal factor L from the L*D*L**H factorization of A. Also, E can be regarded as the superdiagonal of the unit bidiagonal factor U from the U**H *D*U factorization of A. It's giving dual-purpose icon. 🦋

INFO

INFO is INTEGER. Here's the breakdown, bestie:

  • = 0: successful exit. We love to see it. 👑
  • < 0: if INFO = -k, the k-th argument had an illegal value. Not the vibe. 😭
  • > 0: if INFO = k, the leading minor of order k is not positive definite; if k < N, the factorization could not be completed, while if k = N, the factorization was completed, but D(N) <= 0. She's trying her best but no. 💀

Author

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

cpttrs

   subroutine cpttrs (character UPLO, integer N, integer NRHS, real, dimension(
       * ) D, complex, dimension( * ) E, complex, dimension( ldb, * ) B,
       integer LDB, integer INFO)

CPTTRS — okay so this one solves a tridiagonal system. Let's get into it. 💖

Purpose

CPTTRS solves a tridiagonal system of the form

A * X = B

using the factorization A = U**H*D*U or A = L*D*L**H computed by CPTTRF. D is a diagonal matrix specified in the vector D, U (or L) is a unit bidiagonal matrix whose superdiagonal (subdiagonal) is specified in the vector E, and X and B are N by NRHS matrices.

Parameters

  • UPLO — UPLO is CHARACTER*1. Specifies the form of the factorization and whether the vector E is the superdiagonal of the upper bidiagonal factor U or the subdiagonal of the lower bidiagonal factor L.
    • = 'U': A = U**H*D*U, E is the superdiagonal of U
    • = 'L': A = L*D*L**H, E is the subdiagonal of L
  • 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 factorization A = U**H*D*U or A = L*D*L**H.
  • E — E is COMPLEX array, dimension (N-1). If UPLO = 'U', the (n-1) superdiagonal elements of the unit bidiagonal factor U from the factorization A = U**H*D*U. If UPLO = 'L', the (n-1) subdiagonal elements of the unit bidiagonal factor L from the factorization A = L*D*L**H.
  • B — B is COMPLEX array, dimension (LDB,NRHS). On entry, the right hand side vectors B for the system of linear equations. On exit, the solution vectors, 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 = -k, the k-th argument had an illegal value

Author

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

cptts2

   subroutine cptts2 (integer IUPLO, integer N, integer NRHS, real, dimension(
       * ) D, complex, dimension( * ) E, complex, dimension( ldb, * ) B,
       integer LDB)

CPTTS2 solves a tridiagonal system of the form AX=B using the L D LH factorization computed by spttrf. She's the internal workhorse, and she ate. 🔥

Purpose

CPTTS2 solves a tridiagonal system of the form

A * X = B

using the factorization A = U**H*D*U or A = L*D*L**H computed by CPTTRF. D is a diagonal matrix specified in the vector D, U (or L) is a unit bidiagonal matrix whose superdiagonal (subdiagonal) is specified in the vector E, and X and B are N by NRHS matrices.

Parameters

  • IUPLO — IUPLO is INTEGER. Specifies the form of the factorization and whether the vector E is the superdiagonal of the upper bidiagonal factor U or the subdiagonal of the lower bidiagonal factor L.
    • = 1: A = U**H *D*U, E is the superdiagonal of U
    • = 0: A = L*D*L**H, E is the subdiagonal of L
  • 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 factorization A = U**H *D*U or A = L*D*L**H.
  • E — E is COMPLEX array, dimension (N-1). If IUPLO = 1, the (n-1) superdiagonal elements of the unit bidiagonal factor U from the factorization A = U**H*D*U. If IUPLO = 0, the (n-1) subdiagonal elements of the unit bidiagonal factor L from the factorization A = L*D*L**H.
  • B — B is COMPLEX array, dimension (LDB,NRHS). On entry, the right hand side vectors B for the system of linear equations. On exit, the solution vectors, X.
  • LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N).

Author

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

Author

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