labrd(3)

📦
version 3.12.1 · 9 versions
, in arch rolling

labrd — a step in gebrd 💅

okay bestie, this one's from the Library Functions Manual and it's the glow-up page for labrd, aka the little helper that does one iconic step inside gebrd. buckle up 🎀

✨ NAME ✨

labrd — literally what it says: a step in gebrd. 🦋

✨ SYNOPSIS ✨

🌸 Functions 🌸

  • subroutine clabrd (m, n, nb, a, lda, d, e, tauq, taup, x, ldx, y, ldy) — CLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form. 💜
  • subroutine dlabrd (m, n, nb, a, lda, d, e, tauq, taup, x, ldx, y, ldy) — DLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form. 💜
  • subroutine slabrd (m, n, nb, a, lda, d, e, tauq, taup, x, ldx, y, ldy) — SLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form. 💜
  • subroutine zlabrd (m, n, nb, a, lda, d, e, tauq, taup, x, ldx, y, ldy) — ZLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form. 💜

✨ Detailed Description ✨

hii so this page has a whole Detailed Description section and then a Function Documentation section, and that second one is where all the actual tea is ☕ — every subroutine gets its own glow-up down there. 👑

✨ Function Documentation ✨

🦋 subroutine clabrd (integer m, integer n, integer nb, complex, dimension( lda, * ) a, integer lda, real, dimension( * ) d, real, dimension( * ) e, complex, dimension( * ) tauq, complex, dimension( * ) taup, complex, dimension( ldx, * ) x, integer ldx, complex, dimension( ldy, * ) y, integer ldy)

CLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form. no cap, that's the whole gig. 💅

⭐ Purpose:

!>
!> CLABRD reduces the first NB rows and columns of a complex general
!> m by n matrix A to upper or lower real bidiagonal form by a unitary
!> transformation Q**H * A * P, and returns the matrices X and Y which
!> are needed to apply the transformation to the unreduced part of A.
!>
!> If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower
!> bidiagonal form.
!>
!> This is an auxiliary routine called by CGEBRD
!>

like, it's the auxiliary routine — the supporting character that's actually carrying the whole plot 😭 called by CGEBRD, and it hands back X and Y so the untouched part of A can get updated later. bestie behaviour fr fr 🫶

⭐ Parameters

  • M — M is INTEGER, the number of rows in the matrix A. 📏
!>          M is INTEGER
!>          The number of rows in the matrix A.
!>
  • N — N is INTEGER, the number of columns in the matrix A. 📐
!>          N is INTEGER
!>          The number of columns in the matrix A.
!>
  • NB — NB is INTEGER, the number of leading rows and columns of A that we're reducing. 🎯
!>          NB is INTEGER
!>          The number of leading rows and columns of A to be reduced.
!>
  • A — A is COMPLEX array, dimension (LDA,N). on entry it's the m by n general matrix to be reduced; on exit the first NB rows and columns get overwritten and the rest of the array is untouched, periodt. if m >= n, the elements on and below the diagonal in the first NB columns (together with the array TAUQ) represent the unitary matrix Q as a product of elementary reflectors, and the elements above the diagonal in the first NB rows (together with the array TAUP) represent the unitary matrix P as a product of elementary reflectors. if m < n, it flips: elements below the diagonal in the first NB columns with TAUQ give you Q, and elements on and above the diagonal in the first NB rows with TAUP give you P. see Further Details 👀
!>          A is COMPLEX array, dimension (LDA,N)
!>          On entry, the m by n general matrix to be reduced.
!>          On exit, the first NB rows and columns of the matrix are
!>          overwritten; the rest of the array is unchanged.
!>          If m >= n, elements on and below the diagonal in the first NB
!>            columns, with the array TAUQ, represent the unitary
!>            matrix Q as a product of elementary reflectors; and
!>            elements above the diagonal in the first NB rows, with the
!>            array TAUP, represent the unitary matrix P as a product
!>            of elementary reflectors.
!>          If m < n, elements below the diagonal in the first NB
!>            columns, with the array TAUQ, represent the unitary
!>            matrix Q as a product of elementary reflectors, and
!>            elements on and above the diagonal in the first NB rows,
!>            with the array TAUP, represent the unitary matrix P as
!>            a product of elementary reflectors.
!>          See Further Details.
!>
  • LDA — LDA is INTEGER, the leading dimension of the array A. LDA >= max(1,M). 🚨 that bound is not a suggestion bestie.
!>          LDA is INTEGER
!>          The leading dimension of the array A.  LDA >= max(1,M).
!>
  • D — D is REAL array, dimension (NB). the diagonal elements of the first NB rows and columns of the reduced matrix, with D(i) = A(i,i). 💫
!>          D is REAL array, dimension (NB)
!>          The diagonal elements of the first NB rows and columns of
!>          the reduced matrix.  D(i) = A(i,i).
!>
  • E — E is REAL array, dimension (NB). the off-diagonal elements of the first NB rows and columns of the reduced matrix. 🌈
!>          E is REAL array, dimension (NB)
!>          The off-diagonal elements of the first NB rows and columns of
!>          the reduced matrix.
!>
  • TAUQ — TAUQ is COMPLEX array, dimension (NB). the scalar factors of the elementary reflectors that represent the unitary matrix Q. see Further Details. 🔥
!>          TAUQ is COMPLEX array, dimension (NB)
!>          The scalar factors of the elementary reflectors which
!>          represent the unitary matrix Q. See Further Details.
!>
  • TAUP — TAUP is COMPLEX array, dimension (NB). the scalar factors of the elementary reflectors that represent the unitary matrix P. see Further Details. 🔥
!>          TAUP is COMPLEX array, dimension (NB)
!>          The scalar factors of the elementary reflectors which
!>          represent the unitary matrix P. See Further Details.
!>
  • X — X is COMPLEX array, dimension (LDX,NB). the m-by-nb matrix X required to update the unreduced part of A. 🧚
!>          X is COMPLEX array, dimension (LDX,NB)
!>          The m-by-nb matrix X required to update the unreduced part
!>          of A.
!>
  • LDX — LDX is INTEGER, the leading dimension of the array X. LDX >= max(1,M). 💀 don't play with it.
!>          LDX is INTEGER
!>          The leading dimension of the array X. LDX >= max(1,M).
!>
  • Y — Y is COMPLEX array, dimension (LDY,NB). the n-by-nb matrix Y required to update the unreduced part of A. 🍓
!>          Y is COMPLEX array, dimension (LDY,NB)
!>          The n-by-nb matrix Y required to update the unreduced part
!>          of A.
!>
  • LDY — LDY is INTEGER, the leading dimension of the array Y. LDY >= max(1,N). 👑
!>          LDY is INTEGER
!>          The leading dimension of the array Y. LDY >= max(1,N).
!>

⭐ Author

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

⭐ Further Details:

!>
!>  The matrices Q and P are represented as products of elementary
!>  reflectors:
!>
!>     Q = H(1) H(2) . . . H(nb)  and  P = G(1) G(2) . . . G(nb)
!>
!>  Each H(i) and G(i) has the form:
!>
!>     H(i) = I - tauq * v * v**H  and G(i) = I - taup * u * u**H
!>
!>  where tauq and taup are complex scalars, and v and u are complex
!>  vectors.
!>
!>  If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in
!>  A(i:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in
!>  A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
!>
!>  If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in
!>  A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in
!>  A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
!>
!>  The elements of the vectors v and u together form the m-by-nb matrix
!>  V and the nb-by-n matrix U**H which are needed, with X and Y, to apply
!>  the transformation to the unreduced part of the matrix, using a block
!>  update of the form:  A := A - V*Y**H - X*U**H.
!>
!>  The contents of A on exit are illustrated by the following examples
!>  with nb = 2:
!>
!>  m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):
!>
!>    (  1   1   u1  u1  u1 )           (  1   u1  u1  u1  u1  u1 )
!>    (  v1  1   1   u2  u2 )           (  1   1   u2  u2  u2  u2 )
!>    (  v1  v2  a   a   a  )           (  v1  1   a   a   a   a  )
!>    (  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
!>    (  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
!>    (  v1  v2  a   a   a  )
!>
!>  where a denotes an element of the original matrix which is unchanged,
!>  vi denotes an element of the vector defining H(i), and ui an element
!>  of the vector defining G(i).
!>

okay so lowkey the little diagram is the moment — a is an element of the original matrix that's unchanged, vi is an element of the vector defining H(i), and ui is an element of the vector defining G(i). also the block update A := A - V*Y**H - X*U**H is literally the whole reason X and Y exist. iconic. 🌟

Definition at line 210 of file clabrd.f. 💖

🦋 subroutine dlabrd (integer m, integer n, integer nb, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) d, double precision, dimension( * ) e, double precision, dimension( * ) tauq, double precision, dimension( * ) taup, double precision, dimension( ldx, * ) x, integer ldx, double precision, dimension( ldy, * ) y, integer ldy)

DLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form. the real-precision sibling, same energy. 💜

⭐ Purpose:

!>
!> DLABRD reduces the first NB rows and columns of a real general
!> m by n matrix A to upper or lower bidiagonal form by an orthogonal
!> transformation Q**T * A * P, and returns the matrices X and Y which
!> are needed to apply the transformation to the unreduced part of A.
!>
!> If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower
!> bidiagonal form.
!>
!> This is an auxiliary routine called by DGEBRD
!>

note the glow-up from clabrd: this one's orthogonal with Q**T instead of unitary with Q**H, and it's called by DGEBRD. same auxiliary energy, different flavour. 🍓

⭐ Parameters

  • M — M is INTEGER, the number of rows in the matrix A. 📏
!>          M is INTEGER
!>          The number of rows in the matrix A.
!>
  • N — N is INTEGER, the number of columns in the matrix A. 📐
!>          N is INTEGER
!>          The number of columns in the matrix A.
!>
  • NB — NB is INTEGER, the number of leading rows and columns of A to be reduced. 🎯
!>          NB is INTEGER
!>          The number of leading rows and columns of A to be reduced.
!>
  • A — A is DOUBLE PRECISION array, dimension (LDA,N). on entry, the m by n general matrix to be reduced; on exit, the first NB rows and columns are overwritten and the rest of the array is unchanged. if m >= n, the elements on and below the diagonal in the first NB columns, with the array TAUQ, represent the orthogonal matrix Q as a product of elementary reflectors, and the elements above the diagonal in the first NB rows, with the array TAUP, represent the orthogonal matrix P as a product of elementary reflectors. if m < n, the elements below the diagonal in the first NB columns, with the array TAUQ, represent the orthogonal matrix Q as a product of elementary reflectors, and the elements on and above the diagonal in the first NB rows, with the array TAUP, represent the orthogonal matrix P as a product of elementary reflectors. see Further Details 👀
!>          A is DOUBLE PRECISION array, dimension (LDA,N)
!>          On entry, the m by n general matrix to be reduced.
!>          On exit, the first NB rows and columns of the matrix are
!>          overwritten; the rest of the array is unchanged.
!>          If m >= n, elements on and below the diagonal in the first NB
!>            columns, with the array TAUQ, represent the orthogonal
!>            matrix Q as a product of elementary reflectors; and
!>            elements above the diagonal in the first NB rows, with the
!>            array TAUP, represent the orthogonal matrix P as a product
!>            of elementary reflectors.
!>          If m < n, elements below the diagonal in the first NB
!>            columns, with the array TAUQ, represent the orthogonal
!>            matrix Q as a product of elementary reflectors, and
!>            elements on and above the diagonal in the first NB rows,
!>            with the array TAUP, represent the orthogonal matrix P as
!>            a product of elementary reflectors.
!>          See Further Details.
!>
  • LDA — LDA is INTEGER, the leading dimension of the array A. LDA >= max(1,M). 🚨
!>          LDA is INTEGER
!>          The leading dimension of the array A.  LDA >= max(1,M).
!>
  • D — D is DOUBLE PRECISION array, dimension (NB). the diagonal elements of the first NB rows and columns of the reduced matrix, with D(i) = A(i,i). 💫
!>          D is DOUBLE PRECISION array, dimension (NB)
!>          The diagonal elements of the first NB rows and columns of
!>          the reduced matrix.  D(i) = A(i,i).
!>
  • E — E is DOUBLE PRECISION array, dimension (NB). the off-diagonal elements of the first NB rows and columns of the reduced matrix. 🌈
!>          E is DOUBLE PRECISION array, dimension (NB)
!>          The off-diagonal elements of the first NB rows and columns of
!>          the reduced matrix.
!>
  • TAUQ — TAUQ is DOUBLE PRECISION array, dimension (NB). the scalar factors of the elementary reflectors that represent the orthogonal matrix Q. see Further Details. 🔥
!>          TAUQ is DOUBLE PRECISION array, dimension (NB)
!>          The scalar factors of the elementary reflectors which
!>          represent the orthogonal matrix Q. See Further Details.
!>
  • TAUP — TAUP is DOUBLE PRECISION array, dimension (NB). the scalar factors of the elementary reflectors that represent the orthogonal matrix P. see Further Details. 🔥
!>          TAUP is DOUBLE PRECISION array, dimension (NB)
!>          The scalar factors of the elementary reflectors which
!>          represent the orthogonal matrix P. See Further Details.
!>
  • X — X is DOUBLE PRECISION array, dimension (LDX,NB). the m-by-nb matrix X required to update the unreduced part of A. 🧚
!>          X is DOUBLE PRECISION array, dimension (LDX,NB)
!>          The m-by-nb matrix X required to update the unreduced part
!>          of A.
!>
  • LDX — LDX is INTEGER, the leading dimension of the array X... 🫶
!>          LDX is INTEGER
  • LDX — INTEGER. It's the leading dimension of the X array, and you need LDX >= max(1,M). Slay those dimensions! ✨
  • Y — DOUBLE PRECISION array, dimension (LDY,NB). This is the n-by-nb matrix Y that you need to update the unreduced part of A. It's giving “essential helper” energy. 💖
  • LDY — INTEGER. Leading dimension of the Y array, so LDY >= max(1,N). No cap, just facts. 🫶

🌸 Author 🌸

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

💫 Further Details 💫

Okay so the matrices Q and P are represented as products of elementary reflectors — iconic behavior:

Q = H(1) H(2) . . . H(nb)  and  P = G(1) G(2) . . . G(nb)

Each H(i) and G(i) has the form:

H(i) = I - tauq * v * v**T  and G(i) = I - taup * u * u**T

where tauq and taup are real scalars, and v and u are real vectors. (Yes, even in the double precision dlabrd, the text says “real scalars” — we don’t make the rules, we just report them. 💅)

If m >= n, then v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in A(i:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).

If m < n, then v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).

The elements of the vectors v and u together form the m-by-nb matrix V and the nb-by-n matrix U**T which are needed, with X and Y, to apply the transformation to the unreduced part of the matrix, using a block update of the form:

A := A - V*Y**T - X*U**T

The contents of A on exit are illustrated by the following examples with nb = 2:

m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):

(  1   1   u1  u1  u1 )           (  1   u1  u1  u1  u1  u1 )
(  v1  1   1   u2  u2 )           (  1   1   u2  u2  u2  u2 )
(  v1  v2  a   a   a  )           (  v1  1   a   a   a   a  )
(  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
(  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
(  v1  v2  a   a   a  )

where a denotes an element of the original matrix which is unchanged, vi denotes an element of the vector defining H(i), and ui an element of the vector defining G(i).

Definition at line 208 of file dlabrd.f.

🦋 subroutine slabrd 🦋

   subroutine slabrd (integer m, integer n, integer nb, real, dimension( lda, *
       ) a, integer lda, real, dimension( * ) d, real, dimension( * ) e, real,
       dimension( * ) tauq, real, dimension( * ) taup, real, dimension( ldx, *
       ) x, integer ldx, real, dimension( ldy, * ) y, integer ldy)

SLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form. Bestie, it's the real version. ✨

🎯 Purpose

SLABRD reduces the first NB rows and columns of a real general m by n matrix A to upper or lower bidiagonal form by an orthogonal transformation Q**T * A * P, and returns the matrices X and Y which are needed to apply the transformation to the unreduced part of A.

If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower bidiagonal form.

This is an auxiliary routine called by SGEBRD. So like, it's the backstage crew making the main character shine. 👑

📋 Parameters

  • M — INTEGER. The number of rows in the matrix A. Literally the row count, no cap. 📏
  • N — INTEGER. The number of columns in the matrix A. The column count, periodt. 📐
  • NB — INTEGER. The number of leading rows and columns of A to be reduced. So it's how many you're touching. 💅
  • A — REAL array, dimension (LDA,N). On entry, the m-by-n general matrix to be reduced. On exit, the first NB rows and columns of the matrix are overwritten; the rest of the array is unchanged.
    • If m >= n, elements on and below the diagonal in the first NB columns, with the array TAUQ, represent the orthogonal matrix Q as a product of elementary reflectors; and elements above the diagonal in the first NB rows, with the array TAUP, represent the orthogonal matrix P as a product of elementary reflectors.
    • If m < n, elements below the diagonal in the first NB columns, with the array TAUQ, represent the orthogonal matrix Q as a product of elementary reflectors, and elements on and above the diagonal in the first NB rows, with the array TAUP, represent the orthogonal matrix P as a product of elementary reflectors.
    • See Further Details. 🫶
  • LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,M). Keep it valid, bestie. ✨
  • D — REAL array, dimension (NB). The diagonal elements of the first NB rows and columns of the reduced matrix. Specifically D(i) = A(i,i). So it's the main diagonal glow-up. 💖
  • E — REAL array, dimension (NB). The off-diagonal elements of the first NB rows and columns of the reduced matrix. The sidekick energy. 🌸
  • TAUQ — REAL array, dimension (NB). The scalar factors of the elementary reflectors which represent the orthogonal matrix Q. See Further Details. 👑
  • TAUP — REAL array, dimension (NB). The scalar factors of the elementary reflectors which represent the orthogonal matrix P. See Further Details. 👑
  • X — REAL array, dimension (LDX,NB). The m-by-nb matrix X required to update the unreduced part of A. It's giving “essential update matrix”. 🔥
  • LDX — INTEGER. The leading dimension of the array X. LDX >= max(1,M). No shortcuts. 💅
  • Y — REAL array, dimension (LDY,NB). The n-by-nb matrix Y required to update the unreduced part of A. The other half of the update dream team. 🦋
  • LDY — INTEGER. The leading dimension of the array Y. LDY >= max(1,N). Slay those dimensions. ✨

🌟 Author 🌟

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

💫 Further Details 💫

Okay so this is the same reflector situation as before, but let's run it back for slabrd because it's important. The matrices Q and P are represented as products of elementary reflectors:

Q = H(1) H(2) . . . H(nb)  and  P = G(1) G(2) . . . G(nb)

Each H(i) and G(i) has the form:

H(i) = I - tauq * v * v**T  and G(i) = I - taup * u * u**T

where tauq and taup are real scalars, and v and u are real vectors. (Again, the text says “real scalars” — and for slabrd that's literally true. 💅)

If m >= n, then v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in A(i:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).

If m < n, then v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).

The elements of the vectors v and u together form the m-by-nb matrix V and the nb-by-n matrix U**T which are needed, with X and Y, to apply the transformation to the unreduced part of the matrix, using a block update of the form:

A := A - V*Y**T - X*U**T

The contents of A on exit are illustrated by the following examples with nb = 2:

m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):

(  1   1   u1  u1  u1 )           (  1   u1  u1  u1  u1  u1 )
(  v1  1   1   u2  u2 )           (  1   1   u2  u2  u2  u2 )
(  v1  v2  a   a   a  )           (  v1  1   a   a   a   a  )
(  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
(  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
(  v1  v2  a   a   a  )

where a denotes an element of the original matrix which is unchanged, vi denotes an element of the vector defining H(i), and ui an element of the vector defining G(i).

Definition at line 208 of file slabrd.f.

🧚 subroutine zlabrd 🧚

   subroutine zlabrd (integer m, integer n, integer nb, complex*16, dimension(
       lda, * ) a, integer lda, double precision, dimension( * ) d, double
       precision, dimension( * ) e, complex*16, dimension( * ) tauq,
       complex*16, dimension( * ) taup, complex*16, dimension( ldx, * ) x,
       integer ldx, complex*16, dimension( ldy, * ) y, integer ldy)

ZLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form. This is the complex*16 edition, so you know it's fancy. 💜

🎯 Purpose

ZLABRD reduces the first NB rows and columns of a complex general m by n matrix A to upper or lower real bidiagonal form by a unitary transformation Q**H * A * P, and returns the matrices X and Y which are needed to apply the transformation to the unreduced part of A.

If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower bidiagonal form.

This is an auxiliary routine called by ZGEBRD. So it's the backstage baddie for the complex case. 👑

📋 Parameters

  • M — INTEGER. The number of rows in the matrix A. Row count, periodt. 📏
  • N — INTEGER. The number of columns in the matrix A. Column count, no cap. 📐
  • NB — INTEGER. The number of leading rows and columns of A to be reduced. How many you're reducing, bestie. 💅
  • A — COMPLEX*16 array, dimension (LDA,*). (The rest of this parameter's description continues in the next part — stay tuned, it's about to get good! 💖)
  • A — COMPLEX*16 array, dimension (LDA,N). On entry, it’s the m by n general matrix that’s getting reduced. On exit, the first NB rows and columns of the matrix are overwritten; the rest of the array is unchanged. If m >= n, elements on and below the diagonal in the first NB columns, with the array TAUQ, represent the unitary matrix Q as a product of elementary reflectors; and elements above the diagonal in the first NB rows, with the array TAUP, represent the unitary matrix P as a product of elementary reflectors. If m < n, elements below the diagonal in the first NB columns, with the array TAUQ, represent the unitary matrix Q as a product of elementary reflectors, and elements on and above the diagonal in the first NB rows, with the array TAUP, represent the unitary matrix P as a product of elementary reflectors. See Further Details. 🫶
  • LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,M). 🎀
  • D — DOUBLE PRECISION array, dimension (NB). The diagonal elements of the first NB rows and columns of the reduced matrix. D(i) = A(i,i). ✨
  • E — DOUBLE PRECISION array, dimension (NB). The off-diagonal elements of the first NB rows and columns of the reduced matrix. 💫
  • TAUQ — COMPLEX*16 array, dimension (NB). The scalar factors of the elementary reflectors which represent the unitary matrix Q. See Further Details. 💜
  • TAUP — COMPLEX*16 array, dimension (NB). The scalar factors of the elementary reflectors which represent the unitary matrix P. See Further Details. 💖
  • X — COMPLEX*16 array, dimension (LDX,NB). The m-by-nb matrix X required to update the unreduced part of A. 🌸
  • LDX — INTEGER. The leading dimension of the array X. LDX >= max(1,M). 🦋
  • Y — COMPLEX*16 array, dimension (LDY,NB). The n-by-nb matrix Y required to update the unreduced part of A. ⭐
  • LDY — INTEGER. The leading dimension of the array Y. LDY >= max(1,N). 🌈

👑 Author

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

🔎 Further Details:

Okay bestie, here’s the lore: the matrices Q and P are represented as products of elementary reflectors:

Q = H(1) H(2) . . . H(nb)  and  P = G(1) G(2) . . . G(nb)

Each H(i) and G(i) has the form:

H(i) = I - tauq * v * v**H  and G(i) = I - taup * u * u**H

where tauq and taup are complex scalars, and v and u are complex vectors. 💅

If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in A(i:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i). 🔥

If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i). 👑

The elements of the vectors v and u together form the m-by-nb matrix V and the nb-by-n matrix U**H which are needed, with X and Y, to apply the transformation to the unreduced part of the matrix, using a block update of the form: A := A - V*Y**H - X*U**H. 🧚

The contents of A on exit are illustrated by the following examples with nb = 2:

  m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):

    (  1   1   u1  u1  u1 )           (  1   u1  u1  u1  u1  u1 )
    (  v1  1   1   u2  u2 )           (  1   1   u2  u2  u2  u2 )
    (  v1  v2  a   a   a  )           (  v1  1   a   a   a   a  )
    (  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
    (  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
    (  v1  v2  a   a   a  )

where a denotes an element of the original matrix which is unchanged, vi denotes an element of the vector defining H(i), and ui an element of the vector defining G(i). 💖

📌 Definition at line 210 of file zlabrd.f.

👑 Author

Generated automatically by Doxygen for LAPACK from the source code. 💅

That’s the whole page, bestie — signing off from LAPACK Version 3.12.0, page labrd(3). 💖✨

📦 where this page comes from