launhr_col_getrfnp(3)

📦
version 3.12.1 · 9 versions
, in arch rolling

🎀 launhr_col_getrfnp — LU factor without pivoting 🎀

okay bestie so this is part of the LAPACK gang, and the manual page is literally called launhr_col_getrfnp(3). It's giving "niche but iconic" — a whole family of la{un,or}hr_col_getrfnp routines that do LU factor without pivoting. Let's get into it 💅

✨ NAME ✨

launhr_col_getrfnp — la{un,or}hr_col_getrfnp: LU factor without pivoting

fr fr that's the whole name section. Short, sweet, ate.

🦋 SYNOPSIS 🦋

Functions

  • subroutine claunhr_col_getrfnp (m, n, a, lda, d, info) — the CLAUNHR_COL_GETRFNP one 💜
  • subroutine dlaorhr_col_getrfnp (m, n, a, lda, d, info) — the DLAORHR_COL_GETRFNP one 💜
  • subroutine slaorhr_col_getrfnp (m, n, a, lda, d, info) — the SLAORHR_COL_GETRFNP one 💜
  • subroutine zlaunhr_col_getrfnp (m, n, a, lda, d, info) — the ZLAUNHR_COL_GETRFNP one 💜

🌸 Detailed Description 🌸

like, the fine print. keep reading, we get to the actual functions below 🌈

👑 Function Documentation 👑

💖 subroutine claunhr_col_getrfnp (integer m, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( * ) d, integer info)

CLAUNHR_COL_GETRFNP

Purpose:

okay so CLAUNHR_COL_GETRFNP computes the modified LU factorization without pivoting of a complex general M-by-N matrix A. The factorization has the form:

A - S = L * U,

where, and I'm quoting the vibe here:

  • S is a m-by-n diagonal sign matrix with the diagonal D, so that D(i) = S(i,i), 1 <= i <= min(M,N). The diagonal D is constructed as D(i)=-SIGN(A(i,i)), where A(i,i) is the value after performing i-1 steps of Gaussian elimination. This means that the diagonal element at each step of Gaussian elimination is at least one in absolute value — so division-by-zero is not possible during the division by the diagonal element 🫶
  • L is a M-by-N lower triangular matrix with unit diagonal elements (lower trapezoidal if M > N)
  • and U is a M-by-N upper triangular matrix (upper trapezoidal if M < N)

This routine is an auxiliary routine used in the Householder reconstruction routine CUNHR_COL. In CUNHR_COL, this routine is applied to an M-by-N matrix A with orthonormal columns, where each element is bounded by one in absolute value. With the choice of the matrix S above, one can show that the diagonal element at each step of Gaussian elimination is the largest (in absolute value) in the column on or below the diagonal, so that no pivoting is required for numerical stability [1]. no cap 💀

For more details on the Householder reconstruction algorithm, including the modified LU factorization, see [1].

This is the blocked right-looking version of the algorithm, calling Level 3 BLAS to update the submatrix. To factorize a block, this routine calls the recursive routine CLAUNHR_COL_GETRFNP2. recursion girlies rise 🙈

[1] G. Ballard, J. Demmel, L. Grigori, M. Jacquelin, H.D. Nguyen, E. Solomonik, J. Parallel Distrib. Comput., vol. 85, pp. 3-31, 2015.

Parameters

  • M — M is INTEGER. The number of rows of the matrix A. M >= 0.
  • N — N is INTEGER. The number of columns of the matrix A. N >= 0.
  • A — A is COMPLEX array, dimension (LDA,N). On entry, the M-by-N matrix to be factored. On exit, the factors L and U from the factorization A-S=L*U; the unit diagonal elements of L are not stored.
  • LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,M).
  • D — D is COMPLEX array, dimension min(M,N). The diagonal elements of the diagonal M-by-N sign matrix S, D(i) = S(i,i), where 1 <= i <= min(M,N). The elements can be only ( +1.0, 0.0 ) or (-1.0, 0.0 ).
  • 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.

Contributors:

November 2019, Igor Kozachenko, Computer Science Division, University of California, Berkeley

Definition at line 145 of file claunhr_col_getrfnp.f.

💖 subroutine dlaorhr_col_getrfnp (integer m, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) d, integer info)

DLAORHR_COL_GETRFNP

Purpose:

DLAORHR_COL_GETRFNP computes the modified LU factorization without pivoting of a real general M-by-N matrix A. The factorization has the form:

A - S = L * U,

where:

  • S is a m-by-n diagonal sign matrix with the diagonal D, so that D(i) = S(i,i), 1 <= i <= min(M,N). The diagonal D is constructed as D(i)=-SIGN(A(i,i)), where A(i,i) is the value after performing i-1 steps of Gaussian elimination. This means that the diagonal element at each step of Gaussian elimination is at least one in absolute value — so division-by-zero is not possible during the division by the diagonal element 🫶
  • L is a M-by-N lower triangular matrix with unit diagonal elements (lower trapezoidal if M > N)
  • and U is a M-by-N upper triangular matrix (upper trapezoidal if M < N)

This routine is an auxiliary routine used in the Householder reconstruction routine DORHR_COL. In DORHR_COL, this routine is applied to an M-by-N matrix A with orthonormal columns, where each element is bounded by one in absolute value. With the choice of the matrix S above, one can show that the diagonal element at each step of Gaussian elimination is the largest (in absolute value) in the column on or below the diagonal, so that no pivoting is required for numerical stability [1].

For more details on the Householder reconstruction algorithm, including the modified LU factorization, see [1].

This is the blocked right-looking version of the algorithm, calling Level 3 BLAS to update the submatrix. To factorize a block, this routine calls the recursive routine DLAORHR_COL_GETRFNP2. 🌟

[1] G. Ballard, J. Demmel, L. Grigori, M. Jacquelin, H.D. Nguyen, E. Solomonik, J. Parallel Distrib. Comput., vol. 85, pp. 3-31, 2015.

Parameters

  • M — M is INTEGER. The number of rows of the matrix A. M >= 0.
  • N — N is INTEGER. The number of columns of the matrix A. N >= 0.
  • A — A is DOUBLE PRECISION array, dimension (LDA,N). On entry, the M-by-N matrix to be factored. On exit, the factors L and U from the factorization A-S=L*U; the unit diagonal elements of L are not stored.
  • LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,M).
  • D — D is DOUBLE PRECISION array, dimension min(M,N). The diagonal elements of the diagonal M-by-N sign matrix S, D(i) = S(i,i), where 1 <= i <= min(M,N). The elements can be only plus or minus one.
  • 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.

Contributors:

November 2019, Igor Kozachenko, Computer Science Division, University of California, Berkeley

Definition at line 145 of file dlaorhr_col_getrfnp.f.

💖 subroutine slaorhr_col_getrfnp (integer m, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( * ) d, integer info)

SLAORHR_COL_GETRFNP

Purpose:

SLAORHR_COL_GETRFNP computes the modified LU factorization without pivoting of a real general M-by-N matrix A. The factorization has the form:

A - S = L * U,

where:

  • S is a m-by-n diagonal sign matrix with the diagonal D, so that D(i) = S(i,i), 1 <= i <= min(M,N). The diagonal D is constructed as D(i)=-SIGN(A(i,i)), where A(i,i) is the value after performing i-1 steps of Gaussian elimination. This means that the diagonal element at each step of Gaussian elimination is at least one in absolute value — so division-by-zero is not possible during the division by the diagonal element 🫶
  • L is a M-by-N lower triangular matrix with unit diagonal elements (lower trapezoidal if M > N)
  • and U is a M-by-N upper triangular matrix (upper trapezoidal if M < N)

This routine is an auxiliary routine used in the Householder reconstruction routine SORHR_COL. In SORHR_COL, this routine is applied to an M-by-N matrix A with orthonormal columns, where each element is bounded by one in absolute value. With the choice of the matrix S above, one can show that the diagonal element at each step of Gaussian elimination is the largest (in absolute value) in the column on or below the diagonal, so that no pivoting is required for numerical stability [1].

For more details on the Householder reconstruction algorithm, including the modified LU factorization, see [1].

This is the blocked right-looking version of the algorithm, calling Level 3 BLAS to update the submatrix. To factorize a block, this routine calls the recursive routine SLAORHR_COL_GETRFNP2. 🧚

[1] G. Ballard, J. Demmel, L. Grigori, M. Jacquelin, H.D. Nguyen, E. Solomonik, J. Parallel Distrib. Comput., vol. 85, pp. 3-31, 2015.

Parameters

  • M — M is INTEGER. The number of rows of the matrix A. M >= 0.
  • N — N is INTEGER. The number of columns of the matrix A. N >= 0.
  • A —

so like, picking right back up with A 💅 — on entry it's the M-by-N matrix to be factored, and on exit you're left with the factors L and U from the factorization A-S=L*U. The unit diagonal elements of L? Not stored. We simply don't keep them, bestie. 🫶

  • LDA — LDA is INTEGER. It's the leading dimension of the array A, and the vibe check it must pass is LDA >= max(1,M).
  • D — D is REAL array, dimension min(M,N). These are the diagonal elements of that diagonal M-by-N sign matrix S, meaning D(i) = S(i,i), where 1 <= i <= min(M,N). The elements can be only plus or minus one — that's it, that's the whole menu. ✨
  • INFO — INFO is INTEGER, and it's the little status report at the end:
    • = 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.

🎀 Contributors:

November 2019, Igor Kozachenko,
               Computer Science Division,
               University of California, Berkeley
Definition at line 145 of file slaorhr_col_getrfnp.f — iconic, honestly.

🌈 subroutine zlaunhr_col_getrfnp — same energy, but make it complex*16

subroutine zlaunhr_col_getrfnp (integer m, integer n, complex*16, dimension(
    lda, * ) a, integer lda, complex*16, dimension( * ) d, integer info)

ZLAUNHR_COL_GETRFNP — the complex twin, and it ate. 💜

✨ Purpose:

ZLAUNHR_COL_GETRFNP computes the modified LU factorization without pivoting of a complex general M-by-N matrix A. The factorization has the form:

A - S = L * U,

where, okay, let me break it down for you:

  • S is a m-by-n diagonal sign matrix with the diagonal D, so that D(i) = S(i,i), 1 <= i <= min(M,N). The diagonal D is constructed as D(i)=-SIGN(A(i,i)), where A(i,i) is the value after performing i-1 steps of Gaussian elimination. What that means: the diagonal element at each step of Gaussian elimination is at least one in absolute value — so division-by-zero is literally not possible during the division by the diagonal element. slay. 🎀
  • L is a M-by-N lower triangular matrix with unit diagonal elements (lower trapezoidal if M > N);
  • and U is a M-by-N upper triangular matrix (upper trapezoidal if M < N).

This routine is an auxiliary routine used in the Householder reconstruction routine ZUNHR_COL. In ZUNHR_COL, this routine is applied to an M-by-N matrix A with orthonormal columns, where each element is bounded by one in absolute value. With the choice of the matrix S above, one can show that the diagonal element at each step of Gaussian elimination is the largest (in absolute value) in the column on or below the diagonal, so that no pivoting is required for numerical stability [1]. no cap, that's the whole flex. 🔥

For more details on the Householder reconstruction algorithm, including the modified LU factorization, see [1].

This is the blocked right-looking version of the algorithm, calling Level 3 BLAS to update the submatrix. To factorize a block, this routine calls the recursive routine ZLAUNHR_COL_GETRFNP2. recursion staying booked and busy 🧚

[1] ,
    G. Ballard, J. Demmel, L. Grigori, M. Jacquelin, H.D. Nguyen,
    E. Solomonik, J. Parallel Distrib. Comput.,
    vol. 85, pp. 3-31, 2015.

🌸 Parameters

  • M — M is INTEGER, the number of rows of the matrix A. Constraint: M >= 0.
  • N — N is INTEGER, the number of columns of the matrix A. Constraint: N >= 0.
  • A — A is COMPLEX*16 array, dimension (LDA,N). On entry, the M-by-N matrix to be factored. On exit, the factors L and U from the factorization A-S=L*U; the unit diagonal elements of L are not stored. (Again. Still not stored. Never stored. 💅)
  • LDA — LDA is INTEGER, the leading dimension of the array A. Constraint: LDA >= max(1,M).
  • D — D is COMPLEX*16 array, dimension min(M,N). These are the diagonal elements of the diagonal M-by-N sign matrix S, so D(i) = S(i,i), where 1 <= i <= min(M,N). The elements can be only ( +1.0, 0.0 ) or (-1.0, 0.0 ) — complex edition, but still just a sign, bestie. ⭐
  • 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.

🎀 Contributors:

November 2019, Igor Kozachenko,
               Computer Science Division,
               University of California, Berkeley
Definition at line 145 of file zlaunhr_col_getrfnp.f — same line number, different file, we love a consistency moment.

💖 Author

This whole page was generated automatically by Doxygen for LAPACK from the source code. No human hands, just vibes and a doc generator doing its thing. 🤖✨

and that's the page, bestie — everything came straight out of LAPACK, Version 3.12.0, and the footer reads launhr_col_getrfnp(3). go forth and factor responsibly 🫶💜🎀

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