1: #include <petsc.h>
2: #include <petscblaslapack.h>
6: static PetscErrorCode estsv(PetscInt n, PetscReal *r, PetscInt ldr, PetscReal *svmin, PetscReal *z) 7: {
8: PetscBLASInt blas1=1, blasn=n, blasnmi, blasj, blasldr = ldr;
9: PetscInt i,j;
10: PetscReal e,temp,w,wm,ynorm,znorm,s,sm;
13: for (i=0;i<n;i++) {
14: z[i]=0.0;
15: }
16: e = PetscAbs(r[0]);
17: if (e == 0.0) {
18: *svmin = 0.0;
19: z[0] = 1.0;
20: } else {
21: /* Solve R'*y = e */
22: for (i=0;i<n;i++) {
23: /* Scale y. The scaling factor (0.01) reduces the number of scalings */
24: if (z[i] >= 0.0) e =-PetscAbs(e);
25: else e = PetscAbs(e);
27: if (PetscAbs(e - z[i]) > PetscAbs(r[i + ldr*i])) {
28: temp = PetscMin(0.01,PetscAbs(r[i + ldr*i]))/PetscAbs(e-z[i]);
29: PetscStackCallBLAS("BLASscal",BLASscal_(&blasn, &temp, z, &blas1));
30: e = temp*e;
31: }
33: /* Determine the two possible choices of y[i] */
34: if (r[i + ldr*i] == 0.0) {
35: w = wm = 1.0;
36: } else {
37: w = (e - z[i]) / r[i + ldr*i];
38: wm = - (e + z[i]) / r[i + ldr*i];
39: }
41: /* Chose y[i] based on the predicted value of y[j] for j>i */
42: s = PetscAbs(e - z[i]);
43: sm = PetscAbs(e + z[i]);
44: for (j=i+1;j<n;j++) {
45: sm += PetscAbs(z[j] + wm * r[i + ldr*j]);
46: }
47: if (i < n-1) {
48: blasnmi = n-i-1;
49: PetscStackCallBLAS("BLASaxpy",BLASaxpy_(&blasnmi, &w, &r[i + ldr*(i+1)], &blasldr, &z[i+1], &blas1));
50: s += BLASasum_(&blasnmi, &z[i+1], &blas1);
51: }
52: if (s < sm) {
53: temp = wm - w;
54: w = wm;
55: if (i < n-1) {
56: PetscStackCallBLAS("BLASaxpy",BLASaxpy_(&blasnmi, &temp, &r[i + ldr*(i+1)], &blasldr, &z[i+1], &blas1));
57: }
58: }
59: z[i] = w;
60: }
62: ynorm = BLASnrm2_(&blasn, z, &blas1);
64: /* Solve R*z = y */
65: for (j=n-1; j>=0; j--) {
66: /* Scale z */
67: if (PetscAbs(z[j]) > PetscAbs(r[j + ldr*j])) {
68: temp = PetscMin(0.01, PetscAbs(r[j + ldr*j] / z[j]));
69: PetscStackCallBLAS("BLASscal",BLASscal_(&blasn, &temp, z, &blas1));
70: ynorm *=temp;
71: }
72: if (r[j + ldr*j] == 0) {
73: z[j] = 1.0;
74: } else {
75: z[j] = z[j] / r[j + ldr*j];
76: }
77: temp = -z[j];
78: blasj=j;
79: PetscStackCallBLAS("BLASaxpy",BLASaxpy_(&blasj,&temp,&r[0+ldr*j],&blas1,z,&blas1));
80: }
82: /* Compute svmin and normalize z */
83: znorm = 1.0 / BLASnrm2_(&blasn, z, &blas1);
84: *svmin = ynorm*znorm;
85: PetscStackCallBLAS("BLASscal",BLASscal_(&blasn, &znorm, z, &blas1));
86: }
87: return(0);
88: }
90: /*
91: c ***********
92: c
93: c Subroutine dgqt
94: c
95: c Given an n by n symmetric matrix A, an n-vector b, and a
96: c positive number delta, this subroutine determines a vector
97: c x which approximately minimizes the quadratic function
98: c 99: c f(x) = (1/2)*x'*A*x + b'*x
100: c101: c subject to the Euclidean norm constraint102: c103: c norm(x) <= delta.
104: c105: c This subroutine computes an approximation x and a Lagrange106: c multiplier par such that either par is zero and107: c108: c norm(x) <= (1+rtol)*delta,
109: c110: c or par is positive and111: c112: c abs(norm(x) - delta) <= rtol*delta.
113: c
114: c If xsol is the solution to the problem, the approximation x
115: c satisfies116: c117: c f(x) <= ((1 - rtol)**2)*f(xsol)
118: c119: c The subroutine statement is120: c121: c subroutine dgqt(n,a,lda,b,delta,rtol,atol,itmax,122: c par,f,x,info,z,wa1,wa2)123: c
124: c where
125: c
126: c n is an integer variable.
127: c On entry n is the order of A.
128: c On exit n is unchanged.
129: c130: c a is a double precision array of dimension (lda,n).
131: c On entry the full upper triangle of a must contain the
132: c full upper triangle of the symmetric matrix A.
133: c On exit the array contains the matrix A.
134: c
135: c lda is an integer variable.
136: c On entry lda is the leading dimension of the array a.
137: c On exit lda is unchanged.
138: c
139: c b is an double precision array of dimension n.
140: c On entry b specifies the linear term in the quadratic.
141: c On exit b is unchanged.
142: c
143: c delta is a double precision variable.
144: c On entry delta is a bound on the Euclidean norm of x.
145: c On exit delta is unchanged.
146: c
147: c rtol is a double precision variable.
148: c On entry rtol is the relative accuracy desired in the
149: c solution. Convergence occurs if
150: c151: c f(x) <= ((1 - rtol)**2)*f(xsol)
152: c
153: c On exit rtol is unchanged.
154: c
155: c atol is a double precision variable.
156: c On entry atol is the absolute accuracy desired in the
157: c solution. Convergence occurs when
158: c159: c norm(x) <= (1 + rtol)*delta
160: c
161: c max(-f(x),-f(xsol)) <= atol
162: c
163: c On exit atol is unchanged.
164: c
165: c itmax is an integer variable.
166: c On entry itmax specifies the maximum number of iterations.
167: c On exit itmax is unchanged.
168: c
169: c par is a double precision variable.
170: c On entry par is an initial estimate of the Lagrange171: c multiplier for the constraint norm(x) <= delta.
172: c On exit par contains the final estimate of the multiplier.
173: c
174: c f is a double precision variable.
175: c On entry f need not be specified.
176: c On exit f is set to f(x) at the output x.
177: c
178: c x is a double precision array of dimension n.
179: c On entry x need not be specified.
180: c On exit x is set to the final estimate of the solution.
181: c
182: c info is an integer variable.
183: c On entry info need not be specified.
184: c On exit info is set as follows:
185: c
186: c info = 1 The function value f(x) has the relative
187: c accuracy specified by rtol.
188: c
189: c info = 2 The function value f(x) has the absolute
190: c accuracy specified by atol.
191: c
192: c info = 3 Rounding errors prevent further progress.
193: c On exit x is the best available approximation.
194: c
195: c info = 4 Failure to converge after itmax iterations.
196: c On exit x is the best available approximation.
197: c
198: c z is a double precision work array of dimension n.
199: c
200: c wa1 is a double precision work array of dimension n.
201: c
202: c wa2 is a double precision work array of dimension n.
203: c
204: c Subprograms called
205: c
206: c MINPACK-2 ...... destsv
207: c
208: c LAPACK ......... dpotrf
209: c
210: c Level 1 BLAS ... daxpy, dcopy, ddot, dnrm2, dscal
211: c
212: c Level 2 BLAS ... dtrmv, dtrsv
213: c
214: c MINPACK-2 Project. October 1993.
215: c Argonne National Laboratory and University of Minnesota.
216: c Brett M. Averick, Richard Carter, and Jorge J. More'
217: c
218: c ***********
219: */
222: PetscErrorCode gqt(PetscInt n, PetscReal *a, PetscInt lda, PetscReal *b,223: PetscReal delta, PetscReal rtol, PetscReal atol,224: PetscInt itmax, PetscReal *retpar, PetscReal *retf,225: PetscReal *x, PetscInt *retinfo, PetscInt *retits,226: PetscReal *z, PetscReal *wa1, PetscReal *wa2)227: {
229: PetscReal f=0.0,p001=0.001,p5=0.5,minusone=-1,delta2=delta*delta;
230: PetscInt iter, j, rednc,info;
231: PetscBLASInt indef;
232: PetscBLASInt blas1=1, blasn=n, iblas, blaslda = lda,blasldap1=lda+1,blasinfo;
233: PetscReal alpha, anorm, bnorm, parc, parf, parl, pars, par=*retpar,paru, prod, rxnorm, rznorm=0.0, temp, xnorm;
236: iter = 0;
237: parf = 0.0;
238: xnorm = 0.0;
239: rxnorm = 0.0;
240: rednc = 0;
241: for (j=0; j<n; j++) {
242: x[j] = 0.0;
243: z[j] = 0.0;
244: }
246: /* Copy the diagonal and save A in its lower triangle */
247: PetscStackCallBLAS("BLAScopy",BLAScopy_(&blasn,a,&blasldap1, wa1, &blas1));
248: for (j=0;j<n-1;j++) {
249: iblas = n - j - 1;
250: PetscStackCallBLAS("BLAScopy",BLAScopy_(&iblas,&a[j + lda*(j+1)], &blaslda, &a[j+1 + lda*j], &blas1));
251: }
253: /* Calculate the l1-norm of A, the Gershgorin row sums, and the
254: l2-norm of b */
255: anorm = 0.0;
256: for (j=0;j<n;j++) {
257: wa2[j] = BLASasum_(&blasn, &a[0 + lda*j], &blas1);
258: CHKMEMQ;
259: anorm = PetscMax(anorm,wa2[j]);
260: }
261: for (j=0;j<n;j++) {
262: wa2[j] = wa2[j] - PetscAbs(wa1[j]);
263: }
264: bnorm = BLASnrm2_(&blasn,b,&blas1);
265: CHKMEMQ;
266: /* Calculate a lower bound, pars, for the domain of the problem.
267: Also calculate an upper bound, paru, and a lower bound, parl,
268: for the Lagrange multiplier. */
269: pars = parl = paru = -anorm;
270: for (j=0;j<n;j++) {
271: pars = PetscMax(pars, -wa1[j]);
272: parl = PetscMax(parl, wa1[j] + wa2[j]);
273: paru = PetscMax(paru, -wa1[j] + wa2[j]);
274: }
275: parl = PetscMax(bnorm/delta - parl,pars);
276: parl = PetscMax(0.0,parl);
277: paru = PetscMax(0.0, bnorm/delta + paru);
279: /* If the input par lies outside of the interval (parl, paru),
280: set par to the closer endpoint. */
282: par = PetscMax(par,parl);
283: par = PetscMin(par,paru);
285: /* Special case: parl == paru */
286: paru = PetscMax(paru, (1.0 + rtol)*parl);
288: /* Beginning of an iteration */
290: info = 0;
291: for (iter=1;iter<=itmax;iter++) {
292: /* Safeguard par */
293: if (par <= pars && paru > 0) {
294: par = PetscMax(p001, PetscSqrtScalar(parl/paru)) * paru;
295: }
297: /* Copy the lower triangle of A into its upper triangle and
298: compute A + par*I */
300: for (j=0;j<n-1;j++) {
301: iblas = n - j - 1;
302: PetscStackCallBLAS("BLAScopy",BLAScopy_(&iblas,&a[j+1 + j*lda], &blas1,&a[j + (j+1)*lda], &blaslda));
303: }
304: for (j=0;j<n;j++) {
305: a[j + j*lda] = wa1[j] + par;
306: }
308: /* Attempt the Cholesky factorization of A without referencing
309: the lower triangular part. */
310: PetscStackCallBLAS("LAPACKpotrf",LAPACKpotrf_("U",&blasn,a,&blaslda,&indef));
312: /* Case 1: A + par*I is pos. def. */
313: if (indef == 0) {
315: /* Compute an approximate solution x and save the
316: last value of par with A + par*I pos. def. */
318: parf = par;
319: PetscStackCallBLAS("BLAScopy",BLAScopy_(&blasn, b, &blas1, wa2, &blas1));
320: PetscStackCallBLAS("LAPACKtrtrs",LAPACKtrtrs_("U","T","N",&blasn,&blas1,a,&blaslda,wa2,&blasn,&blasinfo));
321: rxnorm = BLASnrm2_(&blasn, wa2, &blas1);
322: PetscStackCallBLAS("LAPACKtrtrs",LAPACKtrtrs_("U","N","N",&blasn,&blas1,a,&blaslda,wa2,&blasn,&blasinfo));
323: PetscStackCallBLAS("BLAScopy",BLAScopy_(&blasn, wa2, &blas1, x, &blas1));
324: PetscStackCallBLAS("BLASscal",BLASscal_(&blasn, &minusone, x, &blas1));
325: xnorm = BLASnrm2_(&blasn, x, &blas1);
326: CHKMEMQ;
328: /* Test for convergence */
329: if (PetscAbs(xnorm - delta) <= rtol*delta ||
330: (par == 0 && xnorm <= (1.0+rtol)*delta)) {
331: info = 1;
332: }
334: /* Compute a direction of negative curvature and use this
335: information to improve pars. */
337: iblas=blasn*blasn;
339: estsv(n,a,lda,&rznorm,z);
340: CHKMEMQ;
341: pars = PetscMax(pars, par-rznorm*rznorm);
343: /* Compute a negative curvature solution of the form
344: x + alpha*z, where norm(x+alpha*z)==delta */
346: rednc = 0;
347: if (xnorm < delta) {
348: /* Compute alpha */
349: prod = BLASdot_(&blasn, z, &blas1, x, &blas1) / delta;
350: temp = (delta - xnorm)*((delta + xnorm)/delta);
351: alpha = temp/(PetscAbs(prod) + PetscSqrtScalar(prod*prod + temp/delta));
352: if (prod >= 0) alpha = PetscAbs(alpha);
353: else alpha =-PetscAbs(alpha);
355: /* Test to decide if the negative curvature step
356: produces a larger reduction than with z=0 */
357: rznorm = PetscAbs(alpha) * rznorm;
358: if ((rznorm*rznorm + par*xnorm*xnorm)/(delta2) <= par) {
359: rednc = 1;
360: }
361: /* Test for convergence */
362: if (p5 * rznorm*rznorm / delta2 <= rtol*(1.0-p5*rtol)*(par + rxnorm*rxnorm/delta2)) {
363: info = 1;
364: } else if (info == 0 && (p5*(par + rxnorm*rxnorm/delta2) <= atol/delta2)) {
365: info = 2;
366: }
367: }
369: /* Compute the Newton correction parc to par. */
370: if (xnorm == 0) {
371: parc = -par;
372: } else {
373: PetscStackCallBLAS("BLAScopy",BLAScopy_(&blasn, x, &blas1, wa2, &blas1));
374: temp = 1.0/xnorm;
375: PetscStackCallBLAS("BLASscal",BLASscal_(&blasn, &temp, wa2, &blas1));
376: PetscStackCallBLAS("LAPACKtrtrs",LAPACKtrtrs_("U","T","N",&blasn, &blas1, a, &blaslda, wa2, &blasn, &blasinfo));
377: temp = BLASnrm2_(&blasn, wa2, &blas1);
378: parc = (xnorm - delta)/(delta*temp*temp);
379: }
381: /* update parl or paru */
382: if (xnorm > delta) {
383: parl = PetscMax(parl, par);
384: } else if (xnorm < delta) {
385: paru = PetscMin(paru, par);
386: }
387: } else {
388: /* Case 2: A + par*I is not pos. def. */
390: /* Use the rank information from the Cholesky
391: decomposition to update par. */
393: if (indef > 1) {
394: /* Restore column indef to A + par*I. */
395: iblas = indef - 1;
396: PetscStackCallBLAS("BLAScopy",BLAScopy_(&iblas,&a[indef-1 + 0*lda],&blaslda,&a[0 + (indef-1)*lda],&blas1));
397: a[indef-1 + (indef-1)*lda] = wa1[indef-1] + par;
399: /* compute parc. */
400: PetscStackCallBLAS("BLAScopy",BLAScopy_(&iblas,&a[0 + (indef-1)*lda], &blas1, wa2, &blas1));
401: PetscStackCallBLAS("LAPACKtrtrs",LAPACKtrtrs_("U","T","N",&iblas,&blas1,a,&blaslda,wa2,&blasn,&blasinfo));
402: PetscStackCallBLAS("BLAScopy",BLAScopy_(&iblas,wa2,&blas1,&a[0 + (indef-1)*lda],&blas1));
403: temp = BLASnrm2_(&iblas,&a[0 + (indef-1)*lda],&blas1);
404: CHKMEMQ;
405: a[indef-1 + (indef-1)*lda] -= temp*temp;
406: PetscStackCallBLAS("LAPACKtrtr",LAPACKtrtrs_("U","N","N",&iblas,&blas1,a,&blaslda,wa2,&blasn,&blasinfo));
407: }
409: wa2[indef-1] = -1.0;
410: iblas = indef;
411: temp = BLASnrm2_(&iblas,wa2,&blas1);
412: parc = - a[indef-1 + (indef-1)*lda]/(temp*temp);
413: pars = PetscMax(pars,par+parc);
415: /* If necessary, increase paru slightly.
416: This is needed because in some exceptional situations
417: paru is the optimal value of par. */
419: paru = PetscMax(paru, (1.0+rtol)*pars);
420: }
422: /* Use pars to update parl */
423: parl = PetscMax(parl,pars);
425: /* Test for converged. */
426: if (info == 0) {
427: if (iter == itmax) info=4;
428: if (paru <= (1.0+p5*rtol)*pars) info=3;
429: if (paru == 0.0) info = 2;
430: }
432: /* If exiting, store the best approximation and restore
433: the upper triangle of A. */
435: if (info != 0) {
436: /* Compute the best current estimates for x and f. */
437: par = parf;
438: f = -p5 * (rxnorm*rxnorm + par*xnorm*xnorm);
439: if (rednc) {
440: f = -p5 * (rxnorm*rxnorm + par*delta*delta - rznorm*rznorm);
441: PetscStackCallBLAS("BLASaxpy",BLASaxpy_(&blasn, &alpha, z, &blas1, x, &blas1));
442: }
443: /* Restore the upper triangle of A */
444: for (j = 0; j<n; j++) {
445: iblas = n - j - 1;
446: PetscStackCallBLAS("BLAScopy",BLAScopy_(&iblas,&a[j+1 + j*lda],&blas1, &a[j + (j+1)*lda],&blaslda));
447: }
448: iblas = lda+1;
449: PetscStackCallBLAS("BLAScopy",BLAScopy_(&blasn,wa1,&blas1,a,&iblas));
450: break;
451: }
452: par = PetscMax(parl,par+parc);
453: }
454: *retpar = par;
455: *retf = f;
456: *retinfo = info;
457: *retits = iter;
458: CHKMEMQ;
459: return(0);
460: }