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Mise à jour de lapack vers la version 3.3.0.
1: SUBROUTINE DGEBD2( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO ) 2: * 3: * -- LAPACK routine (version 3.2) -- 4: * -- LAPACK is a software package provided by Univ. of Tennessee, -- 5: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 6: * November 2006 7: * 8: * .. Scalar Arguments .. 9: INTEGER INFO, LDA, M, N 10: * .. 11: * .. Array Arguments .. 12: DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAUP( * ), 13: $ TAUQ( * ), WORK( * ) 14: * .. 15: * 16: * Purpose 17: * ======= 18: * 19: * DGEBD2 reduces a real general m by n matrix A to upper or lower 20: * bidiagonal form B by an orthogonal transformation: Q' * A * P = B. 21: * 22: * If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal. 23: * 24: * Arguments 25: * ========= 26: * 27: * M (input) INTEGER 28: * The number of rows in the matrix A. M >= 0. 29: * 30: * N (input) INTEGER 31: * The number of columns in the matrix A. N >= 0. 32: * 33: * A (input/output) DOUBLE PRECISION array, dimension (LDA,N) 34: * On entry, the m by n general matrix to be reduced. 35: * On exit, 36: * if m >= n, the diagonal and the first superdiagonal are 37: * overwritten with the upper bidiagonal matrix B; the 38: * elements below the diagonal, with the array TAUQ, represent 39: * the orthogonal matrix Q as a product of elementary 40: * reflectors, and the elements above the first superdiagonal, 41: * with the array TAUP, represent the orthogonal matrix P as 42: * a product of elementary reflectors; 43: * if m < n, the diagonal and the first subdiagonal are 44: * overwritten with the lower bidiagonal matrix B; the 45: * elements below the first subdiagonal, with the array TAUQ, 46: * represent the orthogonal matrix Q as a product of 47: * elementary reflectors, and the elements above the diagonal, 48: * with the array TAUP, represent the orthogonal matrix P as 49: * a product of elementary reflectors. 50: * See Further Details. 51: * 52: * LDA (input) INTEGER 53: * The leading dimension of the array A. LDA >= max(1,M). 54: * 55: * D (output) DOUBLE PRECISION array, dimension (min(M,N)) 56: * The diagonal elements of the bidiagonal matrix B: 57: * D(i) = A(i,i). 58: * 59: * E (output) DOUBLE PRECISION array, dimension (min(M,N)-1) 60: * The off-diagonal elements of the bidiagonal matrix B: 61: * if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1; 62: * if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1. 63: * 64: * TAUQ (output) DOUBLE PRECISION array dimension (min(M,N)) 65: * The scalar factors of the elementary reflectors which 66: * represent the orthogonal matrix Q. See Further Details. 67: * 68: * TAUP (output) DOUBLE PRECISION array, dimension (min(M,N)) 69: * The scalar factors of the elementary reflectors which 70: * represent the orthogonal matrix P. See Further Details. 71: * 72: * WORK (workspace) DOUBLE PRECISION array, dimension (max(M,N)) 73: * 74: * INFO (output) INTEGER 75: * = 0: successful exit. 76: * < 0: if INFO = -i, the i-th argument had an illegal value. 77: * 78: * Further Details 79: * =============== 80: * 81: * The matrices Q and P are represented as products of elementary 82: * reflectors: 83: * 84: * If m >= n, 85: * 86: * Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1) 87: * 88: * Each H(i) and G(i) has the form: 89: * 90: * H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' 91: * 92: * where tauq and taup are real scalars, and v and u are real vectors; 93: * v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(i+1:m,i); 94: * u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(i,i+2:n); 95: * tauq is stored in TAUQ(i) and taup in TAUP(i). 96: * 97: * If m < n, 98: * 99: * Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m) 100: * 101: * Each H(i) and G(i) has the form: 102: * 103: * H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' 104: * 105: * where tauq and taup are real scalars, and v and u are real vectors; 106: * v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i); 107: * u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n); 108: * tauq is stored in TAUQ(i) and taup in TAUP(i). 109: * 110: * The contents of A on exit are illustrated by the following examples: 111: * 112: * m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n): 113: * 114: * ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 ) 115: * ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 ) 116: * ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 ) 117: * ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 ) 118: * ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 ) 119: * ( v1 v2 v3 v4 v5 ) 120: * 121: * where d and e denote diagonal and off-diagonal elements of B, vi 122: * denotes an element of the vector defining H(i), and ui an element of 123: * the vector defining G(i). 124: * 125: * ===================================================================== 126: * 127: * .. Parameters .. 128: DOUBLE PRECISION ZERO, ONE 129: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) 130: * .. 131: * .. Local Scalars .. 132: INTEGER I 133: * .. 134: * .. External Subroutines .. 135: EXTERNAL DLARF, DLARFG, XERBLA 136: * .. 137: * .. Intrinsic Functions .. 138: INTRINSIC MAX, MIN 139: * .. 140: * .. Executable Statements .. 141: * 142: * Test the input parameters 143: * 144: INFO = 0 145: IF( M.LT.0 ) THEN 146: INFO = -1 147: ELSE IF( N.LT.0 ) THEN 148: INFO = -2 149: ELSE IF( LDA.LT.MAX( 1, M ) ) THEN 150: INFO = -4 151: END IF 152: IF( INFO.LT.0 ) THEN 153: CALL XERBLA( 'DGEBD2', -INFO ) 154: RETURN 155: END IF 156: * 157: IF( M.GE.N ) THEN 158: * 159: * Reduce to upper bidiagonal form 160: * 161: DO 10 I = 1, N 162: * 163: * Generate elementary reflector H(i) to annihilate A(i+1:m,i) 164: * 165: CALL DLARFG( M-I+1, A( I, I ), A( MIN( I+1, M ), I ), 1, 166: $ TAUQ( I ) ) 167: D( I ) = A( I, I ) 168: A( I, I ) = ONE 169: * 170: * Apply H(i) to A(i:m,i+1:n) from the left 171: * 172: IF( I.LT.N ) 173: $ CALL DLARF( 'Left', M-I+1, N-I, A( I, I ), 1, TAUQ( I ), 174: $ A( I, I+1 ), LDA, WORK ) 175: A( I, I ) = D( I ) 176: * 177: IF( I.LT.N ) THEN 178: * 179: * Generate elementary reflector G(i) to annihilate 180: * A(i,i+2:n) 181: * 182: CALL DLARFG( N-I, A( I, I+1 ), A( I, MIN( I+2, N ) ), 183: $ LDA, TAUP( I ) ) 184: E( I ) = A( I, I+1 ) 185: A( I, I+1 ) = ONE 186: * 187: * Apply G(i) to A(i+1:m,i+1:n) from the right 188: * 189: CALL DLARF( 'Right', M-I, N-I, A( I, I+1 ), LDA, 190: $ TAUP( I ), A( I+1, I+1 ), LDA, WORK ) 191: A( I, I+1 ) = E( I ) 192: ELSE 193: TAUP( I ) = ZERO 194: END IF 195: 10 CONTINUE 196: ELSE 197: * 198: * Reduce to lower bidiagonal form 199: * 200: DO 20 I = 1, M 201: * 202: * Generate elementary reflector G(i) to annihilate A(i,i+1:n) 203: * 204: CALL DLARFG( N-I+1, A( I, I ), A( I, MIN( I+1, N ) ), LDA, 205: $ TAUP( I ) ) 206: D( I ) = A( I, I ) 207: A( I, I ) = ONE 208: * 209: * Apply G(i) to A(i+1:m,i:n) from the right 210: * 211: IF( I.LT.M ) 212: $ CALL DLARF( 'Right', M-I, N-I+1, A( I, I ), LDA, 213: $ TAUP( I ), A( I+1, I ), LDA, WORK ) 214: A( I, I ) = D( I ) 215: * 216: IF( I.LT.M ) THEN 217: * 218: * Generate elementary reflector H(i) to annihilate 219: * A(i+2:m,i) 220: * 221: CALL DLARFG( M-I, A( I+1, I ), A( MIN( I+2, M ), I ), 1, 222: $ TAUQ( I ) ) 223: E( I ) = A( I+1, I ) 224: A( I+1, I ) = ONE 225: * 226: * Apply H(i) to A(i+1:m,i+1:n) from the left 227: * 228: CALL DLARF( 'Left', M-I, N-I, A( I+1, I ), 1, TAUQ( I ), 229: $ A( I+1, I+1 ), LDA, WORK ) 230: A( I+1, I ) = E( I ) 231: ELSE 232: TAUQ( I ) = ZERO 233: END IF 234: 20 CONTINUE 235: END IF 236: RETURN 237: * 238: * End of DGEBD2 239: * 240: END