Fsolve error: relative norm of the current step is less than max(option​s.StepTole​rance^2,ep​s) = 1.000000e-12

Hello everyone,

I am running the following code, however, I observe an error:

fsolve stopped because the relative norm of the current step, 7.033578e-13, is less than
max(options.StepTolerance^2,eps) = 1.000000e-12. However, the sum of squared function
values, r = 3.409018e+18, exceeds sqrt(options.FunctionTolerance) = 1.000000e-03.

Optimization Metric                                           Options
relative norm(step) =   7.03e-13              max(StepTolerance^2,eps) =   1e-12 (default)
r =   3.41e+18                                 sqrt(FunctionTolerance) =  1.0e-03 (default)

function F = new(x);
kap_dep = 0.0036; 
kap_div = 0.0012; 
thetam = 3.53;
mtgparam = thetam/(thetam-1);
thetad = - 1.46;
depparam = thetad/(thetad-1);
div_bar = 9;   
chi_saver = 0.601;

log_refi_rate = -3.282968022102128;
log_non_am_share = -0.004357084122431;
log_new_loan_size = 0.484804774865286;
log_nu_t = -0.012277614667639;
log_debt = 0.212443935354230;
log_mstar_bank = 0.193437105171249;
mbank = exp(3.204035192501455);
log_y = -0.033219520228313;
log_ph = -0.282925342335558;
log_h_saver = 1.742131415613060;
log_R_new = 0.015649702109572;
log_inflation_factor = -0.007920530545208;
log_Lam_saver_debt = 0.448184815036209;
log_Lam_saver_0_inv = 0.463834545296377;
Rdep = depparam*(exp(log_R_new)-kap_dep);
log_R_dep = log(depparam*(exp(log_R_new)-kap_dep));
log_pay_bank = 3.300154738807201;
log_m_bank = 3.204035192501455;


F(1) = exp(log_refi_rate) * (exp(x(5)) + (1.0-exp(log_non_am_share)))* exp(log_new_loan_size) +exp(log_nu_t)*x(1)*(1.0-exp(log_refi_rate)) + exp(log_refi_rate)*exp(log_nu_t)*exp(log_debt) - x(1);


F(2) = exp(log_mstar_bank)*(exp(x(5)) - exp(log_non_am_share))+ (1.0 - exp(log_refi_rate))*x(2)*exp(log_nu_t)+ exp(log_refi_rate)*exp(log_nu_t)*exp(mbank) - x(2);
 

F(3) = exp(log_y) + exp(x(4)) - del*exp(log_ph)*exp(log_h_saver) + (Rdep+kap_dep)*exp(log_inflation_factor)*x(3) - x(3);
 

F(4) = log(exp(log_inflation_factor)*(exp(log_pay_bank) -exp(log_R_new)*exp(log_m_bank) - exp(log_mstar_bank) + (exp(log_R_new) - exp(log_R_dep) - kap_dep)*chi_saver * x(3)) - kap_div*0.5*(exp(x(4)) - div_bar).^2) - x(4);
 

F(5) = log(mtgparam*(1.0/x(6) + exp(log_non_am_share) - x(7)/x(6))) - x(5); 
 

F(6) = exp(log_Lam_saver_debt)*(x(6)/exp(log_Lam_saver_0_inv)*exp(log_non_am_share)*(1.0 -exp(log_refi_rate)) + 1.0/(1.0 + kap_div*(exp(x(4))-div_bar))) - x(6);
 

F(7) = exp(log_Lam_saver_debt)*(x(7)/exp(log_Lam_saver_0_inv)*exp(log_non_am_share)*(1.0 -exp(log_refi_rate))+exp(log_refi_rate)*exp(log_non_am_share)*x(6)/exp(log_Lam_saver_0_inv)+1.0/(1.0 + kap_div*(exp(x(4))-div_bar))-(1.0/((1.0 + kap_div*(exp(x(4))-div_bar))*exp(log_Lam_saver_0_inv)))*exp(log_R_new)*exp(log_inflation_factor)) - x(7);

Then I run

x0=rand(1,7);
x=fsolve(@new,x0)

I looked at other documentations and tried options. Could you tell me or provide guidance on how to approach this problem? Thanks.

[darova]

What is del?

What about initial conditions?

new(rand(1,7))
ans =
   1.0e+09 *
  Columns 1 through 5
    0.0000    1.8464   -0.0000   -0.0000    0.0000
  Columns 6 through 7
    0.0000    0.0000

[Amina Enkhbold]

Hello thanks for your reply. del = 0.00493.

[darova]

Are you sure soution exists?

[Alex Sha]

Here are solution:

x1: 0.962291323797881
x2: 37480649161.7629
x3: 11.6402263484377
x4: 1.31242747364373
x5: -3.84080956802629
x6: 27.844012685935
x7: 28.2943746019129

Feval:
0
0
1.06581410364015E-14
-2.66453525910038E-15
3.5527136788005E-15
0
-3.5527136788005E-15

[Amina Enkhbold]

Hello Alex Sha, thanks for your comment. How were you able to solve it?

[Alex Sha]

Hi, Amina, the fsolve function is sensitive to initial start-values your guessed, as you see the above results, the right value of x2 is 37480649161.7629, it is far away from the start-value your provided，it will lead to non-convergence. I used an package other than Matlab, with the ability of global optimization, it means guess of initial start-values are no longer required, the code is simple too:

Constant
del = 0.00493,
kap_dep = 0.0036,
kap_div = 0.0012,
thetam = 3.53,
mtgparam = thetam/(thetam-1),
thetad = - 1.46,
depparam = thetad/(thetad-1),
div_bar = 9,
chi_saver = 0.601,
log_refi_rate = -3.282968022102128,
log_non_am_share = -0.004357084122431,
log_new_loan_size = 0.484804774865286,
log_nu_t = -0.012277614667639,
log_debt = 0.212443935354230,
log_mstar_bank = 0.193437105171249,
mbank = exp(3.204035192501455),
log_y = -0.033219520228313,
log_ph = -0.282925342335558,
log_h_saver = 1.742131415613060,
log_R_new = 0.015649702109572,
log_inflation_factor = -0.007920530545208,
log_Lam_saver_debt = 0.448184815036209,
log_Lam_saver_0_inv = 0.463834545296377,
Rdep = depparam*(exp(log_R_new)-kap_dep),
log_R_dep = ln(depparam*(exp(log_R_new)-kap_dep)),
log_pay_bank = 3.300154738807201,
log_m_bank = 3.204035192501455;
Parameter x(7);
Function
exp(log_refi_rate) * (exp(x5) + (1.0-exp(log_non_am_share)))* exp(log_new_loan_size) +exp(log_nu_t)*x1*(1.0-exp(log_refi_rate)) + exp(log_refi_rate)*exp(log_nu_t)*exp(log_debt) - x1;
exp(log_mstar_bank)*(exp(x5) - exp(log_non_am_share))+ (1.0 - exp(log_refi_rate))*x2*exp(log_nu_t)+ exp(log_refi_rate)*exp(log_nu_t)*exp(mbank) - x2;
exp(log_y) + exp(x4) - del*exp(log_ph)*exp(log_h_saver) + (rdep+kap_dep)*exp(log_inflation_factor)*x3 - x3;
ln(exp(log_inflation_factor)*(exp(log_pay_bank) -exp(log_r_new)*exp(log_m_bank) - exp(log_mstar_bank) + (exp(log_r_new) - exp(log_r_dep) - kap_dep)*chi_saver * x3) - kap_div*0.5*(exp(x4) - div_bar).^2) - x4;
ln(mtgparam*(1.0/x6 + exp(log_non_am_share) - x7/x6)) - x5;
exp(log_lam_saver_debt)*(x6/exp(log_lam_saver_0_inv)*exp(log_non_am_share)*(1.0 -exp(log_refi_rate)) + 1.0/(1.0 + kap_div*(exp(x4)-div_bar))) - x6;
exp(log_lam_saver_debt)*(x7/exp(log_lam_saver_0_inv)*exp(log_non_am_share)*(1.0 -exp(log_refi_rate))+exp(log_refi_rate)*exp(log_non_am_share)*x6/exp(log_lam_saver_0_inv)+1.0/(1.0 + kap_div*(exp(x4)-div_bar))-(1.0/((1.0 + kap_div*(exp(x4)-div_bar))*exp(log_lam_saver_0_inv)))*exp(log_r_new)*exp(log_inflation_factor)) - x7;

[Amina Enkhbold]

Thanks for your comment.

[Yulun Wu]

Hello, I have had a very similar problem to you. I am confused with the reply of Alex Sha. How can I install the package mentioned in the reply?

[Walter Roberson]

@Yulun Wu

Alex Sha uses the commercial package at http://www.7d-soft.com/en/

Alex has posted a number of excellent results from the package over the years. I am sometimes able to do slightly better, but it takes me hours to do better than what Alex can do in a few minutes.

However, the software is not inexpensive. I think the price is probably worth it for people who do a lot of optimization... but I would not be able to justify it for myself, for example.

[Yulun Wu]

Thanks for your reply.