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SGEBD2(l)) SGEBD2(l)NAMESGEBD2 - reduce a real general m by n matrix A to upper or lower bidiagonal form B by an orthogonal transformationSYNOPSISSUBROUTINE SGEBD2( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO ) INTEGER INFO, LDA, M, N REAL A( LDA, * ), D( * ), E( * ), TAUP( * ), TAUQ( * ), WORK( * )PURPOSESGEBD2 reduces a real general m by n matrix A to upper or lower bidiagonal form B by an orthogonal transformation: Q' * A * P = B. If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.ARGUMENTSM (input) INTEGER The number of rows in the matrix A. M >= 0. N (input) INTEGER The number of columns in the matrix A. N >= 0. A (input/output) REAL array, dimension (LDA,N) On entry, the m by n general matrix to be reduced. On exit, if m >= n, the diago- nal and the first superdiagonal are overwritten with the upper bidiagonal matrix B; the elements below the diagonal, with the array TAUQ, represent the orthogonal matrix Q as a product of elementary reflectors, and the elements above the first superdiagonal, with the array TAUP, represent the orthogonal matrix P as a product of elementary reflectors; if m < n, the diagonal and the first subdiagonal are overwritten with the lower bidiagonal matrix B; the elements below the first sub- diagonal, with the array TAUQ, represent the orthogonal matrix Q as a product of elementary reflectors, and the elements above the diagonal, with the array TAUP, represent the orthogonal matrix P as a product of elementary reflectors. See Fur- ther Details. LDA (input) INTEGER The leading dimension of the array A. LDA >= max(1,M). D (output) REAL array, dimension (min(M,N)) The diagonal elements of the bidiagonal matrix B: D(i) = A(i,i). E (output) REAL array, dimension (min(M,N)-1) The off-diagonal elements of the bidiagonal matrix B: if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1; if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1. TAUQ (output) REAL array dimension (min(M,N)) The scalar factors of the elementary reflectors which represent the orthogonal matrix Q. See Further Details. TAUP (output) REAL array, dimension (min(M,N)) The scalar factors of the elementary reflectors which represent the orthogonal matrix P. See Further Details. WORK (workspace) REAL array, dimension (max(M,N)) INFO (output) INTEGER = 0: successful exit. < 0: if INFO =, the i-th argument had an illegal value.-iFURTHER DETAILSThe matrices Q and P are represented as products of elementary reflectors: If m >= n, Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1) Each H(i) and G(i) has the form: H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' where tauq and taup are real scalars, and v and u are real vectors; v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(i+1:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(i,i+2:n); tauq is stored in TAUQ(i) and taup in TAUP(i). If m < n, Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m) Each H(i) and G(i) has the form: H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' where tauq and taup are real scalars, and v and u are real vectors; v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i); u(1:i-1) = 0, u(i) = 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). The contents of A on exit are illustrated by the following examples: m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n): ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 ) ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 ) ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 ) ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 ) ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 ) ( v1 v2 v3 v4 v5 ) where d and e denote diagonal and off-diagonal elements of B, vi denotes an element of the vector defining H(i), and ui an element of the vector defining G(i).LAPACK version 3.015 June 2000 SGEBD2(l)

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