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---
:name: slatrd
:md5sum: c95f0cc9aadb9f343ff8ec7c58fd0f69
:category: :subroutine
:arguments:
- uplo:
:type: char
:intent: input
- n:
:type: integer
:intent: input
- nb:
:type: integer
:intent: input
- a:
:type: real
:intent: input/output
:dims:
- lda
- n
- lda:
:type: integer
:intent: input
- e:
:type: real
:intent: output
:dims:
- n-1
- tau:
:type: real
:intent: output
:dims:
- n-1
- w:
:type: real
:intent: output
:dims:
- ldw
- MAX(n,nb)
- ldw:
:type: integer
:intent: input
:substitutions:
ldw: MAX(1,n)
:fortran_help: " SUBROUTINE SLATRD( UPLO, N, NB, A, LDA, E, TAU, W, LDW )\n\n\
* Purpose\n\
* =======\n\
*\n\
* SLATRD reduces NB rows and columns of a real symmetric matrix A to\n\
* symmetric tridiagonal form by an orthogonal similarity\n\
* transformation Q' * A * Q, and returns the matrices V and W which are\n\
* needed to apply the transformation to the unreduced part of A.\n\
*\n\
* If UPLO = 'U', SLATRD reduces the last NB rows and columns of a\n\
* matrix, of which the upper triangle is supplied;\n\
* if UPLO = 'L', SLATRD reduces the first NB rows and columns of a\n\
* matrix, of which the lower triangle is supplied.\n\
*\n\
* This is an auxiliary routine called by SSYTRD.\n\
*\n\n\
* Arguments\n\
* =========\n\
*\n\
* UPLO (input) CHARACTER*1\n\
* Specifies whether the upper or lower triangular part of the\n\
* symmetric matrix A is stored:\n\
* = 'U': Upper triangular\n\
* = 'L': Lower triangular\n\
*\n\
* N (input) INTEGER\n\
* The order of the matrix A.\n\
*\n\
* NB (input) INTEGER\n\
* The number of rows and columns to be reduced.\n\
*\n\
* A (input/output) REAL array, dimension (LDA,N)\n\
* On entry, the symmetric matrix A. If UPLO = 'U', the leading\n\
* n-by-n upper triangular part of A contains the upper\n\
* triangular part of the matrix A, and the strictly lower\n\
* triangular part of A is not referenced. If UPLO = 'L', the\n\
* leading n-by-n lower triangular part of A contains the lower\n\
* triangular part of the matrix A, and the strictly upper\n\
* triangular part of A is not referenced.\n\
* On exit:\n\
* if UPLO = 'U', the last NB columns have been reduced to\n\
* tridiagonal form, with the diagonal elements overwriting\n\
* the diagonal elements of A; the elements above the diagonal\n\
* with the array TAU, represent the orthogonal matrix Q as a\n\
* product of elementary reflectors;\n\
* if UPLO = 'L', the first NB columns have been reduced to\n\
* tridiagonal form, with the diagonal elements overwriting\n\
* the diagonal elements of A; the elements below the diagonal\n\
* with the array TAU, represent the orthogonal matrix Q as a\n\
* product of elementary reflectors.\n\
* See Further Details.\n\
*\n\
* LDA (input) INTEGER\n\
* The leading dimension of the array A. LDA >= (1,N).\n\
*\n\
* E (output) REAL array, dimension (N-1)\n\
* If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal\n\
* elements of the last NB columns of the reduced matrix;\n\
* if UPLO = 'L', E(1:nb) contains the subdiagonal elements of\n\
* the first NB columns of the reduced matrix.\n\
*\n\
* TAU (output) REAL array, dimension (N-1)\n\
* The scalar factors of the elementary reflectors, stored in\n\
* TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'.\n\
* See Further Details.\n\
*\n\
* W (output) REAL array, dimension (LDW,NB)\n\
* The n-by-nb matrix W required to update the unreduced part\n\
* of A.\n\
*\n\
* LDW (input) INTEGER\n\
* The leading dimension of the array W. LDW >= max(1,N).\n\
*\n\n\
* Further Details\n\
* ===============\n\
*\n\
* If UPLO = 'U', the matrix Q is represented as a product of elementary\n\
* reflectors\n\
*\n\
* Q = H(n) H(n-1) . . . H(n-nb+1).\n\
*\n\
* Each H(i) has the form\n\
*\n\
* H(i) = I - tau * v * v'\n\
*\n\
* where tau is a real scalar, and v is a real vector with\n\
* v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i),\n\
* and tau in TAU(i-1).\n\
*\n\
* If UPLO = 'L', the matrix Q is represented as a product of elementary\n\
* reflectors\n\
*\n\
* Q = H(1) H(2) . . . H(nb).\n\
*\n\
* Each H(i) has the form\n\
*\n\
* H(i) = I - tau * v * v'\n\
*\n\
* where tau is a real scalar, and v is a real vector with\n\
* v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),\n\
* and tau in TAU(i).\n\
*\n\
* The elements of the vectors v together form the n-by-nb matrix V\n\
* which is needed, with W, to apply the transformation to the unreduced\n\
* part of the matrix, using a symmetric rank-2k update of the form:\n\
* A := A - V*W' - W*V'.\n\
*\n\
* The contents of A on exit are illustrated by the following examples\n\
* with n = 5 and nb = 2:\n\
*\n\
* if UPLO = 'U': if UPLO = 'L':\n\
*\n\
* ( a a a v4 v5 ) ( d )\n\
* ( a a v4 v5 ) ( 1 d )\n\
* ( a 1 v5 ) ( v1 1 a )\n\
* ( d 1 ) ( v1 v2 a a )\n\
* ( d ) ( v1 v2 a a a )\n\
*\n\
* where d denotes a diagonal element of the reduced matrix, a denotes\n\
* an element of the original matrix that is unchanged, and vi denotes\n\
* an element of the vector defining H(i).\n\
*\n\
* =====================================================================\n\
*\n"
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