bdsqr(3)

📦

version 3.12.1 · 9 versions

3.12.1 · arch rolling (latest)
3.12.0 · opensuse-leap 16.0
3.12.1 · ubuntu stonking
3.12.1 · ubuntu questing
3.12.0 · ubuntu noble
3.12.0 · ubuntu noble
3.12.0 · centos-stream 10, fedora 43, fedora 44
3.12.1 · ubuntu resolute
3.12.1 · debian trixie

🦋 bdsqr(3) — bidiagonal SVD, QR iteration (dqds) 🦋

okay bestie, this is the Library Functions Manual page for bdsqr(3), and it's giving LAPACK bidiagonal-SVD QR-iteration realness. no cap. ✨

💅 NAME 💅

bdsqr — bdsqr: bidiagonal SVD, QR iteration (dqds). periodt. 🫶

✨ SYNOPSIS ✨

🎀 Functions 🎀

subroutine cbdsqr (uplo, n, ncvt, nru, ncc, d, e, vt, ldvt, u, ldu, c,
    ldc, rwork, info)
    CBDSQR
subroutine dbdsqr (uplo, n, ncvt, nru, ncc, d, e, vt, ldvt, u, ldu, c,
    ldc, work, info)
    DBDSQR
subroutine sbdsqr (uplo, n, ncvt, nru, ncc, d, e, vt, ldvt, u, ldu, c,
    ldc, work, info)
    SBDSQR
subroutine zbdsqr (uplo, n, ncvt, nru, ncc, d, e, vt, ldvt, u, ldu, c,
    ldc, rwork, info)
    ZBDSQR

🌸 Detailed Description 🌸

No extra prose here, bestie — it goes straight into the function docs. 💅

💖 Function Documentation 💖

🔥 subroutine cbdsqr 🔥

subroutine cbdsqr (character uplo, integer n, integer ncvt, integer nru,
    integer ncc, real, dimension( * ) d, real, dimension( * ) e, complex,
    dimension( ldvt, * ) vt, integer ldvt, complex, dimension( ldu, * ) u,
    integer ldu, complex, dimension( ldc, * ) c, integer ldc, real,
    dimension( * ) rwork, integer info)

👑 `CBDSQR` 👑

💖 Purpose: 💖

okay so CBDSQR computes the singular values and, optionally, the right and/or left singular vectors from the singular value decomposition (SVD) of a real N-by-N (upper or lower) bidiagonal matrix B using the implicit zero-shift QR algorithm. The SVD of B has the form

B = Q * S * P**H

where S is the diagonal matrix of singular values, Q is an orthogonal matrix of left singular vectors, and P is an orthogonal matrix of right singular vectors. If left singular vectors are requested, this subroutine actually returns U*Q instead of Q, and, if right singular vectors are requested, this subroutine returns PH*VT instead of PH, for given complex input matrices U and VT. When U and VT are the unitary matrices that reduce a general matrix A to bidiagonal form: A = U*B*VT, as computed by CGEBRD, then

A = (U*Q) * S * (P**H*VT)

is the SVD of A. Optionally, the subroutine may also compute Q**H*C for a given complex input matrix C.

See by J. Demmel and W. Kahan, LAPACK Working Note #3 (or SIAM J. Sci. Statist. Comput. vol. 11, no. 5, pp. 873-912, Sept 1990) and by B. Parlett and V. Fernando, Technical Report CPAM-554, Mathematics Department, University of California at Berkeley, July 1992 for a detailed description of the algorithm.

🎀 Parameters 🎀

UPLO — UPLO is CHARACTER*1
- = 'U': B is upper bidiagonal;
- = 'L': B is lower bidiagonal.
N — N is INTEGER. The order of the matrix B. N >= 0.
NCVT — NCVT is INTEGER. The number of columns of the matrix VT. NCVT >= 0.
NRU — NRU is INTEGER. The number of rows of the matrix U. NRU >= 0.
NCC — NCC is INTEGER. The number of columns of the matrix C. NCC >= 0.
D — D is REAL array, dimension (N). On entry, the n diagonal elements of the bidiagonal matrix B. On exit, if INFO=0, the singular values of B in decreasing order.
E — E is REAL array, dimension (N-1). On entry, the N-1 offdiagonal elements of the bidiagonal matrix B. On exit, if INFO = 0, E is destroyed; if INFO > 0, D and E will contain the diagonal and superdiagonal elements of a bidiagonal matrix orthogonally equivalent to the one given as input.
VT — VT is COMPLEX array, dimension (LDVT, NCVT). On entry, an N-by-NCVT matrix VT. On exit, VT is overwritten by P**H * VT. Not referenced if NCVT = 0.
LDVT — LDVT is INTEGER. The leading dimension of the array VT. LDVT >= max(1,N) if NCVT > 0; LDVT >= 1 if NCVT = 0.
U — U is COMPLEX array, dimension (LDU, N). On entry, an NRU-by-N matrix U. On exit, U is overwritten by U * Q. Not referenced if NRU = 0.
LDU — LDU is INTEGER. The leading dimension of the array U. LDU >= max(1,NRU).
C — C is COMPLEX array, dimension (LDC, NCC). On entry, an N-by-NCC matrix C. On exit, C is overwritten by Q**H * C. Not referenced if NCC = 0.
LDC — LDC is INTEGER. The leading dimension of the array C. LDC >= max(1,N) if NCC > 0; LDC >=1 if NCC = 0.
RWORK — RWORK is REAL array, dimension (LRWORK). LRWORK = 4*N, if NCVT = NRU = NCC = 0, and LRWORK = 4*(N-1), otherwise.
INFO — INFO is INTEGER
- = 0: successful exit
- < 0: If INFO = -i, the i-th argument had an illegal value
- > 0: the algorithm did not converge; D and E contain the elements of a bidiagonal matrix which is orthogonally similar to the input matrix B; if INFO = i, i elements of E have not converged to zero.

🧚 Internal Parameters: 🧚

TOLMUL — REAL, default = max(10,min(100,EPS**(-1/8))). TOLMUL controls the convergence criterion of the QR loop. If it is positive, TOLMUL*EPS is the desired relative precision in the computed singular values. If it is negative, abs(TOLMUL*EPS*sigma_max) is the desired absolute accuracy in the computed singular values (corresponds to relative accuracy abs(TOLMUL*EPS) in the largest singular value. abs(TOLMUL) should be between 1 and 1/EPS, and preferably between 10 (for fast convergence) and .1/EPS (for there to be some accuracy in the results). Default is to lose at either one eighth or 2 of the available decimal digits in each computed singular value (whichever is smaller).
MAXITR — INTEGER, default = 6. MAXITR controls the maximum number of passes of the algorithm through its inner loop. The algorithms stops (and so fails to converge) if the number of passes through the inner loop exceeds MAXITR*N**2.

💀 Note: 💀

Bug report from Cezary Dendek.
On November 3rd 2023, the INTEGER variable MAXIT = MAXITR*N**2 is removed since it can overflow pretty easily (for N larger or equal than 18,919). We instead use MAXITDIVN = MAXITR*N.

🌈 Author 🌈

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

⭐ Definition ⭐

Definition at line 233 of file cbdsqr.f.

🔥 subroutine dbdsqr 🔥

subroutine dbdsqr (character uplo, integer n, integer ncvt, integer nru,
    integer ncc, double precision, dimension( * ) d, double precision,
    dimension( * ) e, double precision, dimension( ldvt, * ) vt, integer
    ldvt, double precision, dimension( ldu, * ) u, integer ldu, double
    precision, dimension( ldc, * ) c, integer ldc, double precision,
    dimension( * ) work, integer info)

👑 `DBDSQR` 👑

💖 Purpose: 💖

okay so DBDSQR computes the singular values and, optionally, the right and/or left singular vectors from the singular value decomposition (SVD) of a real N-by-N (upper or lower) bidiagonal matrix B using the implicit zero-shift QR algorithm. The SVD of B has the form

B = Q * S * P**T

where S is the diagonal matrix of singular values, Q is an orthogonal matrix of left singular vectors, and P is an orthogonal matrix of right singular vectors. If left singular vectors are requested, this subroutine actually returns U*Q instead of Q, and, if right singular vectors are requested, this subroutine returns PT*VT instead of PT, for given real input matrices U and VT. When U and VT are the orthogonal matrices that reduce a general matrix A to bidiagonal form: A = U*B*VT, as computed by DGEBRD, then

A = (U*Q) * S * (P**T*VT)

is the SVD of A. Optionally, the subroutine may also compute Q**T*C for a given real input matrix C.

🎀 Parameters 🎀

UPLO — UPLO is CHARACTER*1
- = 'U': B is upper bidiagonal;
- = 'L': B is lower bidiagonal.
N — N is INTEGER. The order of the matrix B. N >= 0.
NCVT — NCVT is INTEGER. The number of columns of the matrix VT. NCVT >= 0.
NRU — NRU is INTEGER. The number of rows of the matrix U. NRU >= 0.
NCC — NCC is INTEGER. The number of columns of the matrix C. NCC >= 0.
D — D is DOUBLE PRECISION array, dimension (N). On entry, the n diagonal elements of the bidiagonal matrix B. On exit, if INFO=0, the singular values of B in decreasing order.
E — E is DOUBLE PRECISION array, dimension (N-1). On entry, the N-1 offdiagonal elements of the bidiagonal matrix B. On exit, if INFO = 0, E is destroyed; if INFO > 0, D and E will contain the diagonal and superdiagonal elements of a bidiagonal matrix orthogonally equivalent to the one given as input.
VT
VT — DOUBLE PRECISION array, dimension (LDVT, NCVT). On entry it's an N-by-NCVT matrix VT. On exit, VT gets overwritten by P**T * VT 💅. And listen — it's not referenced if NCVT = 0, so don't even stress about it then.
LDVT — INTEGER, the leading dimension of the array VT. Rules are: LDVT >= max(1,N) if NCVT > 0; LDVT >= 1 if NCVT = 0. Respect the bounds bestie 🫶
U — DOUBLE PRECISION array, dimension (LDU, N). On entry, an NRU-by-N matrix U. On exit, U is overwritten by U * Q. Not referenced if NRU = 0.
LDU — INTEGER, the leading dimension of the array U. Constraint: LDU >= max(1,NRU). That's it, that's the rule ✨
C — DOUBLE PRECISION array, dimension (LDC, NCC). On entry, an N-by-NCC matrix C. On exit, C is overwritten by Q**T * C. Not referenced if NCC = 0.
LDC — INTEGER, the leading dimension of the array C. Constraint: LDC >= max(1,N) if NCC > 0; LDC >=1 if NCC = 0. Note the little >=1 with no space there — that's canon, we keep it exactly 💀
WORK — DOUBLE PRECISION array, dimension (LWORK). Sizing goes like this:

LWORK = 4*N, if NCVT = NRU = NCC = 0, and
LWORK = 4*(N-1), otherwise

INFO — INTEGER, the vibe check at the end 👑
- = 0: successful exit 🌸
- < 0: If INFO = -i, the i-th argument had an illegal value 🙈
- > 0: buckle up, it branches:
  - if NCVT = NRU = NCC = 0,
    - = 1, a split was marked by a positive value in E
    - = 2, current block of Z not diagonalized after 30*N iterations (in inner while loop)
    - = 3, termination criterion of outer while loop not met (program created more than N unreduced blocks)
  - else NCVT = NRU = NCC = 0, the algorithm did not converge; D and E contain the elements of a bidiagonal matrix which is orthogonally similar to the input matrix B; if INFO = i, i elements of E have not converged to zero. No cap, that's a real "it didn't cook" moment 😭

🔧 Internal Parameters

TOLMUL — DOUBLE PRECISION, default = max(10,min(100,EPS**(-1/8))). It controls the convergence criterion of the QR loop. Okay so:
- If it's positive, TOLMUL*EPS is the desired relative precision in the computed singular values.
- If it's negative, abs(TOLMUL*EPS*sigma_max) is the desired absolute accuracy in the computed singular values (corresponds to relative accuracy abs(TOLMUL*EPS) in the largest singular value).
- abs(TOLMUL) should be between 1 and 1/EPS, and preferably between 10 (for fast convergence) and .1/EPS (for there to be some accuracy in the results).
- Default behaviour: lose either one eighth or 2 of the available decimal digits in each computed singular value (whichever is smaller). Lowkey a trade, but it's the documented one 🤷‍♀️
MAXITR — INTEGER, default = 6. Controls the maximum number of passes of the algorithm through its inner loop. The algorithms stops (and so fails to converge) if the number of passes through the inner loop exceeds MAXITR*N**2. Bestie, that's the hard cap 🔥

📌 Note

Bug report from Cezary Dendek. On March 23rd 2017, the INTEGER variable MAXIT = MAXITR*N**2 is removed since it can overflow pretty easily (for N larger or equal than 18,919). We instead use MAXITDIVN = MAXITR*N. Fixed it, slayed it, periodt 💅

👑 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 241 of file dbdsqr.f.

🌈 `sbdsqr` — the `REAL` version, bestie

subroutine sbdsqr (character uplo, integer n, integer ncvt, integer nru,
    integer ncc, real, dimension( * ) d, real, dimension( * ) e, real,
    dimension( ldvt, * ) vt, integer ldvt, real, dimension( ldu, * ) u,
    integer ldu, real, dimension( ldc, * ) c, integer ldc, real, dimension(
    * ) work, integer info)

SBDSQR

🎯 Purpose

SBDSQR computes the singular values and, optionally, the right and/or left singular vectors from the singular value decomposition (SVD) of a real N-by-N (upper or lower) bidiagonal matrix B using the implicit zero-shift QR algorithm. The SVD of B has the form

B = Q * S * P**T

where S is the diagonal matrix of singular values, Q is an orthogonal matrix of left singular vectors, and P is an orthogonal matrix of right singular vectors. If left singular vectors are requested, this subroutine actually returns U*Q instead of Q, and, if right singular vectors are requested, this subroutine returns P**T*VT instead of P**T, for given real input matrices U and VT. When U and VT are the orthogonal matrices that reduce a general matrix A to bidiagonal form: A = U*B*VT, as computed by SGEBRD, then

A = (U*Q) * S * (P**T*VT)

is the SVD of A. Optionally, the subroutine may also compute Q**T*C for a given real input matrix C. It's giving "one subroutine, so many optional side quests" ✨

✨ Parameters

UPLO — CHARACTER*1.
- = 'U': B is upper bidiagonal;
- = 'L': B is lower bidiagonal.
N — INTEGER, the order of the matrix B. N >= 0.
NCVT — INTEGER, the number of columns of the matrix VT. NCVT >= 0.
NRU — INTEGER, the number of rows of the matrix U. NRU >= 0.
NCC — INTEGER, the number of columns of the matrix C. NCC >= 0.
D — REAL array, dimension (N). On entry, the n diagonal elements of the bidiagonal matrix B. On exit, if INFO=0, the singular values of B in decreasing order. Sorted, bestie — decreasing, as it should be 💅
E — REAL array, dimension (N-1). On entry, the N-1 offdiagonal elements of the bidiagonal matrix B. On exit, if INFO = 0, E is destroyed; if INFO > 0, D and E will contain the diagonal and superdiagonal elements of a bidiagonal matrix orthogonally equivalent to the one given as input. Destroyed. Gone. That's not a typo 😭
VT — REAL array, dimension (LDVT, NCVT). On entry, an N-by-NCVT matrix VT. On exit, VT is overwritten by P**T * VT. Not referenced if NCVT = 0.
LDVT — INTEGER, the leading dimension of the array VT. LDVT >= max(1,N) if NCVT > 0; LDVT >= 1 if NCVT = 0.
U — REAL array, dimension (LDU, N). On entry, an NRU-by-N matrix U. On exit, U is overwritten by U * Q. Not referenced if NRU = 0.
LDU — INTEGER, the leading dimension of the array U. LDU >= max(1,NRU).
C — REAL array, dimension (LDC, NCC). On entry, an N-by-NCC matrix C. On exit, C is overwritten by Q**T * C. Not referenced if NCC = 0.
LDC — INTEGER, the leading dimension of the array C. LDC >= max(1,N) if NCC > 0; LDC >=1 if NCC = 0.
WORK — REAL array, dimension (LWORK).

LWORK = 4*N, if NCVT = NRU = NCC = 0, and
LWORK = 4*(N-1), otherwise

INFO — INTEGER, same energy as the double version, here we go:
- = 0: successful exit 🌸
- < 0: If INFO = -i, the i-th argument had an illegal value 🙈
- > 0:
  - if NCVT = NRU = NCC = 0,
    - = 1, a split was marked by a positive value in E
    - = 2, current block of Z not diagonalized after 30*N iterations (in inner while loop)
    - = 3, termination criterion of outer while loop not met (program created more than N unreduced blocks)
  - else NCVT = NRU = NCC = 0, the algorithm did not converge; D and E contain the elements of a bidiagonal matrix which is orthogonally similar to the input matrix B; if INFO = i, i elements of E have not converged to zero.

🔧 Internal Parameters

okay, picking up the internal-parameter tea right where it left off — these are the knobs for that first bidiagonal situation:

TOLMUL — REAL, default = max(10,min(100,EPS**(-1/8))). She controls the convergence criterion of the QR loop. If it’s positive, TOLMUL*EPS is the desired relative precision in the computed singular values. If it’s negative, abs(TOLMUL*EPS*sigma_max) is the desired absolute accuracy in the computed singular values (this corresponds to relative accuracy abs(TOLMUL*EPS) in the largest singular value). abs(TOLMUL) should be between 1 and 1/EPS, and preferably between 10 (for fast convergence) and .1/EPS (so there’s still some accuracy in the results). Default is to lose at either one eighth or 2 of the available decimal digits in each computed singular value (whichever is smaller). <span>💅</span>
MAXITR — INTEGER, default = 6. MAXITR controls the maximum number of passes of the algorithm through its inner loop. The algorithm stops (and so fails to converge) if the number of passes through the inner loop exceeds MAXITR*N**2.

📝 Note 📝

Bug report from Cezary Dendek. On March 23rd 2017, the INTEGER variable MAXIT = MAXITR*N**2 is removed since it can overflow pretty easily (for N larger or equal than 18,919). We instead use MAXITDIVN = MAXITR*N. Slay for not overflowing, honestly.

💖 Author 💖

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

Definition at line 240 of file sbdsqr.f.

👑 ZBDSQR 👑

Here’s the zbdsqr glow-up, signature first:

subroutine zbdsqr (character uplo, integer n, integer ncvt, integer nru,
    integer ncc, double precision, dimension( * ) d, double precision,
    dimension( * ) e, complex*16, dimension( ldvt, * ) vt, integer ldvt,
    complex*16, dimension( ldu, * ) u, integer ldu, complex*16, dimension(
    ldc, * ) c, integer ldc, double precision, dimension( * ) rwork, integer
    info)

🎯 Purpose 🎯

ZBDSQR computes the singular values and, optionally, the right and/or left singular vectors from the singular value decomposition (SVD) of a real N-by-N (upper or lower) bidiagonal matrix B using the implicit zero-shift QR algorithm. The SVD of B has the form

B = Q * S * P**H

where S is the diagonal matrix of singular values, Q is an orthogonal matrix of left singular vectors, and P is an orthogonal matrix of right singular vectors. If left singular vectors are requested, this subroutine actually returns U*Q instead of Q, and, if right singular vectors are requested, this subroutine returns P**H*VT instead of P**H, for given complex input matrices U and VT. When U and VT are the unitary matrices that reduce a general matrix A to bidiagonal form: A = U*B*VT, as computed by ZGEBRD, then

A = (U*Q) * S * (P**H*VT)

is the SVD of A. Optionally, the subroutine may also compute Q**H*C for a given complex input matrix C. It’s giving full SVD main-character energy.

See by J. Demmel and W. Kahan, LAPACK Working Note #3 (or SIAM J. Sci. Statist. Comput. vol. 11, no. 5, pp. 873-912, Sept 1990) and by B. Parlett and V. Fernando, Technical Report CPAM-554, Mathematics Department, University of California at Berkeley, July 1992 for a detailed description of the algorithm. <butterfly>🦋</butterfly>

🧩 Parameters 🧩

UPLO — UPLO is CHARACTER*1; = 'U': B is upper bidiagonal; = 'L': B is lower bidiagonal.
N — N is INTEGER; the order of the matrix B. N >= 0.
NCVT — NCVT is INTEGER; the number of columns of the matrix VT. NCVT >= 0.
NRU — NRU is INTEGER; the number of rows of the matrix U. NRU >= 0.
NCC — NCC is INTEGER; the number of columns of the matrix C. NCC >= 0.
D — D is DOUBLE PRECISION array, dimension (N). On entry, the N diagonal elements of the bidiagonal matrix B. On exit, if INFO=0, the singular values of B in decreasing order.
E — E is DOUBLE PRECISION array, dimension (N-1). On entry, the N-1 offdiagonal elements of the bidiagonal matrix B. On exit, if INFO = 0, E is destroyed; if INFO > 0, D and E will contain the diagonal and superdiagonal elements of a bidiagonal matrix orthogonally equivalent to the one given as input.
VT — VT is COMPLEX*16 array, dimension (LDVT, NCVT). On entry, an N-by-NCVT matrix VT. On exit, VT is overwritten by P**H * VT. Not referenced if NCVT = 0.
LDVT — LDVT is INTEGER; the leading dimension of the array VT. LDVT >= max(1,N) if NCVT > 0; LDVT >= 1 if NCVT = 0.
U — U is COMPLEX*16 array, dimension (LDU, N). On entry, an NRU-by-N matrix U. On exit, U is overwritten by U * Q. Not referenced if NRU = 0.
LDU — LDU is INTEGER; the leading dimension of the array U. LDU >= max(1,NRU).
C — C is COMPLEX*16 array, dimension (LDC, NCC). On entry, an N-by-NCC matrix C. On exit, C is overwritten by Q**H * C. Not referenced if NCC = 0.
LDC — LDC is INTEGER; the leading dimension of the array C. LDC >= max(1,N) if NCC > 0; LDC >= 1 if NCC = 0.
RWORK — RWORK is DOUBLE PRECISION array, dimension (LRWORK). LRWORK = 4*N, if NCVT = NRU = NCC = 0, and LRWORK = 4*(N-1), otherwise.
INFO — INFO is INTEGER; = 0: successful exit; < 0: if INFO = -i, the i-th argument had an illegal value; > 0: the algorithm did not converge; D and E contain the elements of a bidiagonal matrix which is orthogonally similar to the input matrix B; if INFO = i, i elements of E have not converged to zero. So yeah, INFO > 0 is the “bestie, it did not converge” moment.

🔧 Internal Parameters 🔧

TOLMUL — DOUBLE PRECISION, default = max(10,min(100,EPS**(-1/8))). TOLMUL controls the convergence criterion of the QR loop. If it is positive, TOLMUL*EPS is the desired relative precision in the computed singular values. If it is negative, abs(TOLMUL*EPS*sigma_max) is the desired absolute accuracy in the computed singular values (corresponds to relative accuracy abs(TOLMUL*EPS) in the largest singular value). abs(TOLMUL) should be between 1 and 1/EPS, and preferably between 10 (for fast convergence) and .1/EPS (for there to be some accuracy in the results). Default is to lose at either one eighth or 2 of the available decimal digits in each computed singular value (whichever is smaller).
MAXITR — INTEGER, default = 6. MAXITR controls the maximum number of passes of the algorithm through its inner loop. The algorithm stops (and so fails to converge) if the number of passes through the inner loop exceeds MAXITR*N**2.

📝 Note 📝

Bug report from Cezary Dendek. On November 3rd 2023, the INTEGER variable MAXIT = MAXITR*N**2 is removed since it can overflow pretty easily (for N larger or equal than 18,919). We instead use MAXITDIVN = MAXITR*N. Same overflow fix, new date, still iconic.

💖 Author 💖

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

Definition at line 233 of file zbdsqr.f.

🧚 Author 🧚

Generated automatically by Doxygen for LAPACK from the source code.

✨ That’s the whole bdsqr(3) page, bestie — source/version: LAPACK Version 3.12.0; page: bdsqr(3). Now go forth and slay those singular values! 💖👑

📦 where this page comes from

arch rolling: lapack-doc 3.12.1-2 · custom
💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash-1017 on 2026-11-04: a language model's rewrite of the original, under its terms

bdsqr(3)

🦋 bdsqr(3) — bidiagonal SVD, QR iteration (dqds) 🦋

💅 NAME 💅

✨ SYNOPSIS ✨

🎀 Functions 🎀

🌸 Detailed Description 🌸

💖 Function Documentation 💖

🔥 subroutine cbdsqr 🔥

👑 CBDSQR 👑

💖 Purpose: 💖

🎀 Parameters 🎀

🧚 Internal Parameters: 🧚

💀 Note: 💀

🌈 Author 🌈

⭐ Definition ⭐

🔥 subroutine dbdsqr 🔥

👑 DBDSQR 👑

💖 Purpose: 💖

🎀 Parameters 🎀

🔧 Internal Parameters

📌 Note

👑 Author

🌈 sbdsqr — the REAL version, bestie

🎯 Purpose

✨ Parameters

🔧 Internal Parameters

📝 Note 📝

💖 Author 💖

👑 ZBDSQR 👑

🎯 Purpose 🎯

🧩 Parameters 🧩

🔧 Internal Parameters 🔧

📝 Note 📝

💖 Author 💖

🧚 Author 🧚

👑 `CBDSQR` 👑

👑 `DBDSQR` 👑

🌈 `sbdsqr` — the `REAL` version, bestie