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LAPACK 3.3.1
Linear Algebra PACKage
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LAPACK 3.3.1
Linear Algebra PACKage
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00001 SUBROUTINE SLABRD( M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y, 00002 $ LDY ) 00003 * 00004 * -- LAPACK auxiliary routine (version 3.3.1) -- 00005 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00006 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00007 * -- April 2011 -- 00008 * 00009 * .. Scalar Arguments .. 00010 INTEGER LDA, LDX, LDY, M, N, NB 00011 * .. 00012 * .. Array Arguments .. 00013 REAL A( LDA, * ), D( * ), E( * ), TAUP( * ), 00014 $ TAUQ( * ), X( LDX, * ), Y( LDY, * ) 00015 * .. 00016 * 00017 * Purpose 00018 * ======= 00019 * 00020 * SLABRD reduces the first NB rows and columns of a real general 00021 * m by n matrix A to upper or lower bidiagonal form by an orthogonal 00022 * transformation Q**T * A * P, and returns the matrices X and Y which 00023 * are needed to apply the transformation to the unreduced part of A. 00024 * 00025 * If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower 00026 * bidiagonal form. 00027 * 00028 * This is an auxiliary routine called by SGEBRD 00029 * 00030 * Arguments 00031 * ========= 00032 * 00033 * M (input) INTEGER 00034 * The number of rows in the matrix A. 00035 * 00036 * N (input) INTEGER 00037 * The number of columns in the matrix A. 00038 * 00039 * NB (input) INTEGER 00040 * The number of leading rows and columns of A to be reduced. 00041 * 00042 * A (input/output) REAL array, dimension (LDA,N) 00043 * On entry, the m by n general matrix to be reduced. 00044 * On exit, the first NB rows and columns of the matrix are 00045 * overwritten; the rest of the array is unchanged. 00046 * If m >= n, elements on and below the diagonal in the first NB 00047 * columns, with the array TAUQ, represent the orthogonal 00048 * matrix Q as a product of elementary reflectors; and 00049 * elements above the diagonal in the first NB rows, with the 00050 * array TAUP, represent the orthogonal matrix P as a product 00051 * of elementary reflectors. 00052 * If m < n, elements below the diagonal in the first NB 00053 * columns, with the array TAUQ, represent the orthogonal 00054 * matrix Q as a product of elementary reflectors, and 00055 * elements on and above the diagonal in the first NB rows, 00056 * with the array TAUP, represent the orthogonal matrix P as 00057 * a product of elementary reflectors. 00058 * See Further Details. 00059 * 00060 * LDA (input) INTEGER 00061 * The leading dimension of the array A. LDA >= max(1,M). 00062 * 00063 * D (output) REAL array, dimension (NB) 00064 * The diagonal elements of the first NB rows and columns of 00065 * the reduced matrix. D(i) = A(i,i). 00066 * 00067 * E (output) REAL array, dimension (NB) 00068 * The off-diagonal elements of the first NB rows and columns of 00069 * the reduced matrix. 00070 * 00071 * TAUQ (output) REAL array dimension (NB) 00072 * The scalar factors of the elementary reflectors which 00073 * represent the orthogonal matrix Q. See Further Details. 00074 * 00075 * TAUP (output) REAL array, dimension (NB) 00076 * The scalar factors of the elementary reflectors which 00077 * represent the orthogonal matrix P. See Further Details. 00078 * 00079 * X (output) REAL array, dimension (LDX,NB) 00080 * The m-by-nb matrix X required to update the unreduced part 00081 * of A. 00082 * 00083 * LDX (input) INTEGER 00084 * The leading dimension of the array X. LDX >= max(1,M). 00085 * 00086 * Y (output) REAL array, dimension (LDY,NB) 00087 * The n-by-nb matrix Y required to update the unreduced part 00088 * of A. 00089 * 00090 * LDY (input) INTEGER 00091 * The leading dimension of the array Y. LDY >= max(1,N). 00092 * 00093 * Further Details 00094 * =============== 00095 * 00096 * The matrices Q and P are represented as products of elementary 00097 * reflectors: 00098 * 00099 * Q = H(1) H(2) . . . H(nb) and P = G(1) G(2) . . . G(nb) 00100 * 00101 * Each H(i) and G(i) has the form: 00102 * 00103 * H(i) = I - tauq * v * v**T and G(i) = I - taup * u * u**T 00104 * 00105 * where tauq and taup are real scalars, and v and u are real vectors. 00106 * 00107 * If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in 00108 * A(i:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in 00109 * A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i). 00110 * 00111 * If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in 00112 * A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in 00113 * A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i). 00114 * 00115 * The elements of the vectors v and u together form the m-by-nb matrix 00116 * V and the nb-by-n matrix U**T which are needed, with X and Y, to apply 00117 * the transformation to the unreduced part of the matrix, using a block 00118 * update of the form: A := A - V*Y**T - X*U**T. 00119 * 00120 * The contents of A on exit are illustrated by the following examples 00121 * with nb = 2: 00122 * 00123 * m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n): 00124 * 00125 * ( 1 1 u1 u1 u1 ) ( 1 u1 u1 u1 u1 u1 ) 00126 * ( v1 1 1 u2 u2 ) ( 1 1 u2 u2 u2 u2 ) 00127 * ( v1 v2 a a a ) ( v1 1 a a a a ) 00128 * ( v1 v2 a a a ) ( v1 v2 a a a a ) 00129 * ( v1 v2 a a a ) ( v1 v2 a a a a ) 00130 * ( v1 v2 a a a ) 00131 * 00132 * where a denotes an element of the original matrix which is unchanged, 00133 * vi denotes an element of the vector defining H(i), and ui an element 00134 * of the vector defining G(i). 00135 * 00136 * ===================================================================== 00137 * 00138 * .. Parameters .. 00139 REAL ZERO, ONE 00140 PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0 ) 00141 * .. 00142 * .. Local Scalars .. 00143 INTEGER I 00144 * .. 00145 * .. External Subroutines .. 00146 EXTERNAL SGEMV, SLARFG, SSCAL 00147 * .. 00148 * .. Intrinsic Functions .. 00149 INTRINSIC MIN 00150 * .. 00151 * .. Executable Statements .. 00152 * 00153 * Quick return if possible 00154 * 00155 IF( M.LE.0 .OR. N.LE.0 ) 00156 $ RETURN 00157 * 00158 IF( M.GE.N ) THEN 00159 * 00160 * Reduce to upper bidiagonal form 00161 * 00162 DO 10 I = 1, NB 00163 * 00164 * Update A(i:m,i) 00165 * 00166 CALL SGEMV( 'No transpose', M-I+1, I-1, -ONE, A( I, 1 ), 00167 $ LDA, Y( I, 1 ), LDY, ONE, A( I, I ), 1 ) 00168 CALL SGEMV( 'No transpose', M-I+1, I-1, -ONE, X( I, 1 ), 00169 $ LDX, A( 1, I ), 1, ONE, A( I, I ), 1 ) 00170 * 00171 * Generate reflection Q(i) to annihilate A(i+1:m,i) 00172 * 00173 CALL SLARFG( M-I+1, A( I, I ), A( MIN( I+1, M ), I ), 1, 00174 $ TAUQ( I ) ) 00175 D( I ) = A( I, I ) 00176 IF( I.LT.N ) THEN 00177 A( I, I ) = ONE 00178 * 00179 * Compute Y(i+1:n,i) 00180 * 00181 CALL SGEMV( 'Transpose', M-I+1, N-I, ONE, A( I, I+1 ), 00182 $ LDA, A( I, I ), 1, ZERO, Y( I+1, I ), 1 ) 00183 CALL SGEMV( 'Transpose', M-I+1, I-1, ONE, A( I, 1 ), LDA, 00184 $ A( I, I ), 1, ZERO, Y( 1, I ), 1 ) 00185 CALL SGEMV( 'No transpose', N-I, I-1, -ONE, Y( I+1, 1 ), 00186 $ LDY, Y( 1, I ), 1, ONE, Y( I+1, I ), 1 ) 00187 CALL SGEMV( 'Transpose', M-I+1, I-1, ONE, X( I, 1 ), LDX, 00188 $ A( I, I ), 1, ZERO, Y( 1, I ), 1 ) 00189 CALL SGEMV( 'Transpose', I-1, N-I, -ONE, A( 1, I+1 ), 00190 $ LDA, Y( 1, I ), 1, ONE, Y( I+1, I ), 1 ) 00191 CALL SSCAL( N-I, TAUQ( I ), Y( I+1, I ), 1 ) 00192 * 00193 * Update A(i,i+1:n) 00194 * 00195 CALL SGEMV( 'No transpose', N-I, I, -ONE, Y( I+1, 1 ), 00196 $ LDY, A( I, 1 ), LDA, ONE, A( I, I+1 ), LDA ) 00197 CALL SGEMV( 'Transpose', I-1, N-I, -ONE, A( 1, I+1 ), 00198 $ LDA, X( I, 1 ), LDX, ONE, A( I, I+1 ), LDA ) 00199 * 00200 * Generate reflection P(i) to annihilate A(i,i+2:n) 00201 * 00202 CALL SLARFG( N-I, A( I, I+1 ), A( I, MIN( I+2, N ) ), 00203 $ LDA, TAUP( I ) ) 00204 E( I ) = A( I, I+1 ) 00205 A( I, I+1 ) = ONE 00206 * 00207 * Compute X(i+1:m,i) 00208 * 00209 CALL SGEMV( 'No transpose', M-I, N-I, ONE, A( I+1, I+1 ), 00210 $ LDA, A( I, I+1 ), LDA, ZERO, X( I+1, I ), 1 ) 00211 CALL SGEMV( 'Transpose', N-I, I, ONE, Y( I+1, 1 ), LDY, 00212 $ A( I, I+1 ), LDA, ZERO, X( 1, I ), 1 ) 00213 CALL SGEMV( 'No transpose', M-I, I, -ONE, A( I+1, 1 ), 00214 $ LDA, X( 1, I ), 1, ONE, X( I+1, I ), 1 ) 00215 CALL SGEMV( 'No transpose', I-1, N-I, ONE, A( 1, I+1 ), 00216 $ LDA, A( I, I+1 ), LDA, ZERO, X( 1, I ), 1 ) 00217 CALL SGEMV( 'No transpose', M-I, I-1, -ONE, X( I+1, 1 ), 00218 $ LDX, X( 1, I ), 1, ONE, X( I+1, I ), 1 ) 00219 CALL SSCAL( M-I, TAUP( I ), X( I+1, I ), 1 ) 00220 END IF 00221 10 CONTINUE 00222 ELSE 00223 * 00224 * Reduce to lower bidiagonal form 00225 * 00226 DO 20 I = 1, NB 00227 * 00228 * Update A(i,i:n) 00229 * 00230 CALL SGEMV( 'No transpose', N-I+1, I-1, -ONE, Y( I, 1 ), 00231 $ LDY, A( I, 1 ), LDA, ONE, A( I, I ), LDA ) 00232 CALL SGEMV( 'Transpose', I-1, N-I+1, -ONE, A( 1, I ), LDA, 00233 $ X( I, 1 ), LDX, ONE, A( I, I ), LDA ) 00234 * 00235 * Generate reflection P(i) to annihilate A(i,i+1:n) 00236 * 00237 CALL SLARFG( N-I+1, A( I, I ), A( I, MIN( I+1, N ) ), LDA, 00238 $ TAUP( I ) ) 00239 D( I ) = A( I, I ) 00240 IF( I.LT.M ) THEN 00241 A( I, I ) = ONE 00242 * 00243 * Compute X(i+1:m,i) 00244 * 00245 CALL SGEMV( 'No transpose', M-I, N-I+1, ONE, A( I+1, I ), 00246 $ LDA, A( I, I ), LDA, ZERO, X( I+1, I ), 1 ) 00247 CALL SGEMV( 'Transpose', N-I+1, I-1, ONE, Y( I, 1 ), LDY, 00248 $ A( I, I ), LDA, ZERO, X( 1, I ), 1 ) 00249 CALL SGEMV( 'No transpose', M-I, I-1, -ONE, A( I+1, 1 ), 00250 $ LDA, X( 1, I ), 1, ONE, X( I+1, I ), 1 ) 00251 CALL SGEMV( 'No transpose', I-1, N-I+1, ONE, A( 1, I ), 00252 $ LDA, A( I, I ), LDA, ZERO, X( 1, I ), 1 ) 00253 CALL SGEMV( 'No transpose', M-I, I-1, -ONE, X( I+1, 1 ), 00254 $ LDX, X( 1, I ), 1, ONE, X( I+1, I ), 1 ) 00255 CALL SSCAL( M-I, TAUP( I ), X( I+1, I ), 1 ) 00256 * 00257 * Update A(i+1:m,i) 00258 * 00259 CALL SGEMV( 'No transpose', M-I, I-1, -ONE, A( I+1, 1 ), 00260 $ LDA, Y( I, 1 ), LDY, ONE, A( I+1, I ), 1 ) 00261 CALL SGEMV( 'No transpose', M-I, I, -ONE, X( I+1, 1 ), 00262 $ LDX, A( 1, I ), 1, ONE, A( I+1, I ), 1 ) 00263 * 00264 * Generate reflection Q(i) to annihilate A(i+2:m,i) 00265 * 00266 CALL SLARFG( M-I, A( I+1, I ), A( MIN( I+2, M ), I ), 1, 00267 $ TAUQ( I ) ) 00268 E( I ) = A( I+1, I ) 00269 A( I+1, I ) = ONE 00270 * 00271 * Compute Y(i+1:n,i) 00272 * 00273 CALL SGEMV( 'Transpose', M-I, N-I, ONE, A( I+1, I+1 ), 00274 $ LDA, A( I+1, I ), 1, ZERO, Y( I+1, I ), 1 ) 00275 CALL SGEMV( 'Transpose', M-I, I-1, ONE, A( I+1, 1 ), LDA, 00276 $ A( I+1, I ), 1, ZERO, Y( 1, I ), 1 ) 00277 CALL SGEMV( 'No transpose', N-I, I-1, -ONE, Y( I+1, 1 ), 00278 $ LDY, Y( 1, I ), 1, ONE, Y( I+1, I ), 1 ) 00279 CALL SGEMV( 'Transpose', M-I, I, ONE, X( I+1, 1 ), LDX, 00280 $ A( I+1, I ), 1, ZERO, Y( 1, I ), 1 ) 00281 CALL SGEMV( 'Transpose', I, N-I, -ONE, A( 1, I+1 ), LDA, 00282 $ Y( 1, I ), 1, ONE, Y( I+1, I ), 1 ) 00283 CALL SSCAL( N-I, TAUQ( I ), Y( I+1, I ), 1 ) 00284 END IF 00285 20 CONTINUE 00286 END IF 00287 RETURN 00288 * 00289 * End of SLABRD 00290 * 00291 END
1.7.3 
