title: nls singular gradient matrix - fit parameters in integral's upper limits
body: <p>I am trying to make a <code>nls</code> fit for a little bit complicated expression that includes two integrals with two of the fit parameters in their upper limits.</p>
  
  <p>I got the error</p>
  
  <blockquote>
    <p>"Error in nlsModel(formula, mf, start, wts) :  singular gradient
    matrix at initial parameter estimates".</p>
  </blockquote>
  
  <p>I have searched already in the previous answers, but didn't help. The parameters initialization seem to be ok, I have tried to change the parameters but none work. If my function has just one integral everything works very nicely, but when adding a second integral term just got the error. I don't believe the function is over-parametrized, as I have performed other fits with much more parameters and they worked. Below I have wrote a list with some data.</p>
  
  <p>The minimal example is the following:</p>
  
  <pre><code>integrand &lt;- function(X) {
    return(X^4/(2*sinh(X/2))^2)
  }
  
  fitting = function(T1, T2, N, D, x){
    int1 = integrate(integrand, lower=0, upper = T1)$value
    int2 = integrate(integrand, lower=0, upper = T2)$value
    return(N*(D/x)^2*(exp(D/x)/(1+exp(D/x))^2
  )+(448.956*(x/T1)^3*int1)+(299.304*(x/T2)^3*int2))
  }
  fit = nls(y ~ fitting(T1, T2, N, D, x),
  start=list(T1=400,T2=200,N=0.01,D=2))
  </code></pre>
  
  <p>------>For reference, the fit that worked is the following:</p>
  
  <pre><code>integrand &lt;- function(X) {
    return(X^4/(2*sinh(X/2))^2)
  }
  fitting = function(T1, N, D, x){
    int = integrate(integrand, lower=0, upper = T1)$value
    return(N*(D/x)^2*(exp(D/x)/(1+exp(D/x))^2 )+(748.26)*(x/T1)^3*int)
  }
  fit = nls(y ~ fitting(T1 , N, D, x), start=list(T1=400,N=0.01,D=2))
  </code></pre>
  
  <p>------->Data to illustrate the problem:</p>
  
  <pre><code>dat&lt;- read.table(text="x       y
  0.38813 0.0198
  0.79465 0.02206
  1.40744 0.01676
  1.81532 0.01538
  2.23105 0.01513
  2.64864 0.01547
  3.05933 0.01706
  3.47302 0.01852
  3.88791 0.02074
  4.26301 0.0256
  4.67607 0.03028
  5.08172 0.03507
  5.48327 0.04283
  5.88947 0.05017
  6.2988  0.05953
  6.7022  0.07185
  7.10933 0.08598
  7.51924 0.0998
  7.92674 0.12022
  8.3354  0.1423
  8.7384  0.16382
  9.14656 0.19114
  9.55062 0.22218
  9.95591 0.25542", header=TRUE)
  </code></pre>
  
  <p>I cannot figure out what happen. I need to perform this fit for three integral components, but even for two I have this problem. I appreciate so much your help. Thank you.</p>