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---
:name: clabrd
:md5sum: 6ef5300c03ff0c0ef1c02f7f2da050d5
:category: :subroutine
:arguments:
- m:
:type: integer
:intent: input
- n:
:type: integer
:intent: input
- nb:
:type: integer
:intent: input
- a:
:type: complex
:intent: input/output
:dims:
- lda
- n
- lda:
:type: integer
:intent: input
- d:
:type: real
:intent: output
:dims:
- MAX(1,nb)
- e:
:type: real
:intent: output
:dims:
- MAX(1,nb)
- tauq:
:type: complex
:intent: output
:dims:
- MAX(1,nb)
- taup:
:type: complex
:intent: output
:dims:
- MAX(1,nb)
- x:
:type: complex
:intent: output
:dims:
- ldx
- MAX(1,nb)
- ldx:
:type: integer
:intent: input
- y:
:type: complex
:intent: output
:dims:
- ldy
- MAX(1,nb)
- ldy:
:type: integer
:intent: input
:substitutions:
ldx: MAX(1,m)
ldy: MAX(1,n)
:fortran_help: " SUBROUTINE CLABRD( M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y, LDY )\n\n\
* Purpose\n\
* =======\n\
*\n\
* CLABRD reduces the first NB rows and columns of a complex general\n\
* m by n matrix A to upper or lower real bidiagonal form by a unitary\n\
* transformation Q' * A * P, and returns the matrices X and Y which\n\
* are needed to apply the transformation to the unreduced part of A.\n\
*\n\
* If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower\n\
* bidiagonal form.\n\
*\n\
* This is an auxiliary routine called by CGEBRD\n\
*\n\n\
* Arguments\n\
* =========\n\
*\n\
* M (input) INTEGER\n\
* The number of rows in the matrix A.\n\
*\n\
* N (input) INTEGER\n\
* The number of columns in the matrix A.\n\
*\n\
* NB (input) INTEGER\n\
* The number of leading rows and columns of A to be reduced.\n\
*\n\
* A (input/output) COMPLEX array, dimension (LDA,N)\n\
* On entry, the m by n general matrix to be reduced.\n\
* On exit, the first NB rows and columns of the matrix are\n\
* overwritten; the rest of the array is unchanged.\n\
* If m >= n, elements on and below the diagonal in the first NB\n\
* columns, with the array TAUQ, represent the unitary\n\
* matrix Q as a product of elementary reflectors; and\n\
* elements above the diagonal in the first NB rows, with the\n\
* array TAUP, represent the unitary matrix P as a product\n\
* of elementary reflectors.\n\
* If m < n, elements below the diagonal in the first NB\n\
* columns, with the array TAUQ, represent the unitary\n\
* matrix Q as a product of elementary reflectors, and\n\
* elements on and above the diagonal in the first NB rows,\n\
* with the array TAUP, represent the unitary matrix P as\n\
* a product of elementary reflectors.\n\
* See Further Details.\n\
*\n\
* LDA (input) INTEGER\n\
* The leading dimension of the array A. LDA >= max(1,M).\n\
*\n\
* D (output) REAL array, dimension (NB)\n\
* The diagonal elements of the first NB rows and columns of\n\
* the reduced matrix. D(i) = A(i,i).\n\
*\n\
* E (output) REAL array, dimension (NB)\n\
* The off-diagonal elements of the first NB rows and columns of\n\
* the reduced matrix.\n\
*\n\
* TAUQ (output) COMPLEX array dimension (NB)\n\
* The scalar factors of the elementary reflectors which\n\
* represent the unitary matrix Q. See Further Details.\n\
*\n\
* TAUP (output) COMPLEX array, dimension (NB)\n\
* The scalar factors of the elementary reflectors which\n\
* represent the unitary matrix P. See Further Details.\n\
*\n\
* X (output) COMPLEX array, dimension (LDX,NB)\n\
* The m-by-nb matrix X required to update the unreduced part\n\
* of A.\n\
*\n\
* LDX (input) INTEGER\n\
* The leading dimension of the array X. LDX >= max(1,M).\n\
*\n\
* Y (output) COMPLEX array, dimension (LDY,NB)\n\
* The n-by-nb matrix Y required to update the unreduced part\n\
* of A.\n\
*\n\
* LDY (input) INTEGER\n\
* The leading dimension of the array Y. LDY >= max(1,N).\n\
*\n\n\
* Further Details\n\
* ===============\n\
*\n\
* The matrices Q and P are represented as products of elementary\n\
* reflectors:\n\
*\n\
* Q = H(1) H(2) . . . H(nb) and P = G(1) G(2) . . . G(nb)\n\
*\n\
* Each H(i) and G(i) has the form:\n\
*\n\
* H(i) = I - tauq * v * v' and G(i) = I - taup * u * u'\n\
*\n\
* where tauq and taup are complex scalars, and v and u are complex\n\
* vectors.\n\
*\n\
* If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in\n\
* A(i:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in\n\
* A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).\n\
*\n\
* If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in\n\
* A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in\n\
* A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).\n\
*\n\
* The elements of the vectors v and u together form the m-by-nb matrix\n\
* V and the nb-by-n matrix U' which are needed, with X and Y, to apply\n\
* the transformation to the unreduced part of the matrix, using a block\n\
* update of the form: A := A - V*Y' - X*U'.\n\
*\n\
* The contents of A on exit are illustrated by the following examples\n\
* with nb = 2:\n\
*\n\
* m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):\n\
*\n\
* ( 1 1 u1 u1 u1 ) ( 1 u1 u1 u1 u1 u1 )\n\
* ( v1 1 1 u2 u2 ) ( 1 1 u2 u2 u2 u2 )\n\
* ( v1 v2 a a a ) ( v1 1 a a a a )\n\
* ( v1 v2 a a a ) ( v1 v2 a a a a )\n\
* ( v1 v2 a a a ) ( v1 v2 a a a a )\n\
* ( v1 v2 a a a )\n\
*\n\
* where a denotes an element of the original matrix which is unchanged,\n\
* vi denotes an element of the vector defining H(i), and ui an element\n\
* of the vector defining G(i).\n\
*\n\
* =====================================================================\n\
*\n"
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