00001 SUBROUTINE DGEBD2( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO ) 00002 * 00003 * -- LAPACK routine (version 3.2) -- 00004 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00005 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00006 * November 2006 00007 * 00008 * .. Scalar Arguments .. 00009 INTEGER INFO, LDA, M, N 00010 * .. 00011 * .. Array Arguments .. 00012 DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAUP( * ), 00013 $ TAUQ( * ), WORK( * ) 00014 * .. 00015 * 00016 * Purpose 00017 * ======= 00018 * 00019 * DGEBD2 reduces a real general m by n matrix A to upper or lower 00020 * bidiagonal form B by an orthogonal transformation: Q' * A * P = B. 00021 * 00022 * If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal. 00023 * 00024 * Arguments 00025 * ========= 00026 * 00027 * M (input) INTEGER 00028 * The number of rows in the matrix A. M >= 0. 00029 * 00030 * N (input) INTEGER 00031 * The number of columns in the matrix A. N >= 0. 00032 * 00033 * A (input/output) DOUBLE PRECISION array, dimension (LDA,N) 00034 * On entry, the m by n general matrix to be reduced. 00035 * On exit, 00036 * if m >= n, the diagonal and the first superdiagonal are 00037 * overwritten with the upper bidiagonal matrix B; the 00038 * elements below the diagonal, with the array TAUQ, represent 00039 * the orthogonal matrix Q as a product of elementary 00040 * reflectors, and the elements above the first superdiagonal, 00041 * with the array TAUP, represent the orthogonal matrix P as 00042 * a product of elementary reflectors; 00043 * if m < n, the diagonal and the first subdiagonal are 00044 * overwritten with the lower bidiagonal matrix B; the 00045 * elements below the first subdiagonal, with the array TAUQ, 00046 * represent the orthogonal matrix Q as a product of 00047 * elementary reflectors, and the elements above the diagonal, 00048 * with the array TAUP, represent the orthogonal matrix P as 00049 * a product of elementary reflectors. 00050 * See Further Details. 00051 * 00052 * LDA (input) INTEGER 00053 * The leading dimension of the array A. LDA >= max(1,M). 00054 * 00055 * D (output) DOUBLE PRECISION array, dimension (min(M,N)) 00056 * The diagonal elements of the bidiagonal matrix B: 00057 * D(i) = A(i,i). 00058 * 00059 * E (output) DOUBLE PRECISION array, dimension (min(M,N)-1) 00060 * The off-diagonal elements of the bidiagonal matrix B: 00061 * if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1; 00062 * if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1. 00063 * 00064 * TAUQ (output) DOUBLE PRECISION array dimension (min(M,N)) 00065 * The scalar factors of the elementary reflectors which 00066 * represent the orthogonal matrix Q. See Further Details. 00067 * 00068 * TAUP (output) DOUBLE PRECISION array, dimension (min(M,N)) 00069 * The scalar factors of the elementary reflectors which 00070 * represent the orthogonal matrix P. See Further Details. 00071 * 00072 * WORK (workspace) DOUBLE PRECISION array, dimension (max(M,N)) 00073 * 00074 * INFO (output) INTEGER 00075 * = 0: successful exit. 00076 * < 0: if INFO = -i, the i-th argument had an illegal value. 00077 * 00078 * Further Details 00079 * =============== 00080 * 00081 * The matrices Q and P are represented as products of elementary 00082 * reflectors: 00083 * 00084 * If m >= n, 00085 * 00086 * Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1) 00087 * 00088 * Each H(i) and G(i) has the form: 00089 * 00090 * H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' 00091 * 00092 * where tauq and taup are real scalars, and v and u are real vectors; 00093 * v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(i+1:m,i); 00094 * u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(i,i+2:n); 00095 * tauq is stored in TAUQ(i) and taup in TAUP(i). 00096 * 00097 * If m < n, 00098 * 00099 * Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m) 00100 * 00101 * Each H(i) and G(i) has the form: 00102 * 00103 * H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' 00104 * 00105 * where tauq and taup are real scalars, and v and u are real vectors; 00106 * v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i); 00107 * u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n); 00108 * tauq is stored in TAUQ(i) and taup in TAUP(i). 00109 * 00110 * The contents of A on exit are illustrated by the following examples: 00111 * 00112 * m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n): 00113 * 00114 * ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 ) 00115 * ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 ) 00116 * ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 ) 00117 * ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 ) 00118 * ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 ) 00119 * ( v1 v2 v3 v4 v5 ) 00120 * 00121 * where d and e denote diagonal and off-diagonal elements of B, vi 00122 * denotes an element of the vector defining H(i), and ui an element of 00123 * the vector defining G(i). 00124 * 00125 * ===================================================================== 00126 * 00127 * .. Parameters .. 00128 DOUBLE PRECISION ZERO, ONE 00129 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) 00130 * .. 00131 * .. Local Scalars .. 00132 INTEGER I 00133 * .. 00134 * .. External Subroutines .. 00135 EXTERNAL DLARF, DLARFG, XERBLA 00136 * .. 00137 * .. Intrinsic Functions .. 00138 INTRINSIC MAX, MIN 00139 * .. 00140 * .. Executable Statements .. 00141 * 00142 * Test the input parameters 00143 * 00144 INFO = 0 00145 IF( M.LT.0 ) THEN 00146 INFO = -1 00147 ELSE IF( N.LT.0 ) THEN 00148 INFO = -2 00149 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN 00150 INFO = -4 00151 END IF 00152 IF( INFO.LT.0 ) THEN 00153 CALL XERBLA( 'DGEBD2', -INFO ) 00154 RETURN 00155 END IF 00156 * 00157 IF( M.GE.N ) THEN 00158 * 00159 * Reduce to upper bidiagonal form 00160 * 00161 DO 10 I = 1, N 00162 * 00163 * Generate elementary reflector H(i) to annihilate A(i+1:m,i) 00164 * 00165 CALL DLARFG( M-I+1, A( I, I ), A( MIN( I+1, M ), I ), 1, 00166 $ TAUQ( I ) ) 00167 D( I ) = A( I, I ) 00168 A( I, I ) = ONE 00169 * 00170 * Apply H(i) to A(i:m,i+1:n) from the left 00171 * 00172 IF( I.LT.N ) 00173 $ CALL DLARF( 'Left', M-I+1, N-I, A( I, I ), 1, TAUQ( I ), 00174 $ A( I, I+1 ), LDA, WORK ) 00175 A( I, I ) = D( I ) 00176 * 00177 IF( I.LT.N ) THEN 00178 * 00179 * Generate elementary reflector G(i) to annihilate 00180 * A(i,i+2:n) 00181 * 00182 CALL DLARFG( N-I, A( I, I+1 ), A( I, MIN( I+2, N ) ), 00183 $ LDA, TAUP( I ) ) 00184 E( I ) = A( I, I+1 ) 00185 A( I, I+1 ) = ONE 00186 * 00187 * Apply G(i) to A(i+1:m,i+1:n) from the right 00188 * 00189 CALL DLARF( 'Right', M-I, N-I, A( I, I+1 ), LDA, 00190 $ TAUP( I ), A( I+1, I+1 ), LDA, WORK ) 00191 A( I, I+1 ) = E( I ) 00192 ELSE 00193 TAUP( I ) = ZERO 00194 END IF 00195 10 CONTINUE 00196 ELSE 00197 * 00198 * Reduce to lower bidiagonal form 00199 * 00200 DO 20 I = 1, M 00201 * 00202 * Generate elementary reflector G(i) to annihilate A(i,i+1:n) 00203 * 00204 CALL DLARFG( N-I+1, A( I, I ), A( I, MIN( I+1, N ) ), LDA, 00205 $ TAUP( I ) ) 00206 D( I ) = A( I, I ) 00207 A( I, I ) = ONE 00208 * 00209 * Apply G(i) to A(i+1:m,i:n) from the right 00210 * 00211 IF( I.LT.M ) 00212 $ CALL DLARF( 'Right', M-I, N-I+1, A( I, I ), LDA, 00213 $ TAUP( I ), A( I+1, I ), LDA, WORK ) 00214 A( I, I ) = D( I ) 00215 * 00216 IF( I.LT.M ) THEN 00217 * 00218 * Generate elementary reflector H(i) to annihilate 00219 * A(i+2:m,i) 00220 * 00221 CALL DLARFG( M-I, A( I+1, I ), A( MIN( I+2, M ), I ), 1, 00222 $ TAUQ( I ) ) 00223 E( I ) = A( I+1, I ) 00224 A( I+1, I ) = ONE 00225 * 00226 * Apply H(i) to A(i+1:m,i+1:n) from the left 00227 * 00228 CALL DLARF( 'Left', M-I, N-I, A( I+1, I ), 1, TAUQ( I ), 00229 $ A( I+1, I+1 ), LDA, WORK ) 00230 A( I+1, I ) = E( I ) 00231 ELSE 00232 TAUQ( I ) = ZERO 00233 END IF 00234 20 CONTINUE 00235 END IF 00236 RETURN 00237 * 00238 * End of DGEBD2 00239 * 00240 END
1.7.2